Finite Fields and Their Applications 93 (2024) 102316 Contents lists available at ScienceDirect Finite Fields and Their Applications journal homepage: www.elsevier.com/locate/ffa Partial permutation decoding and PD-sets for Zps -linear generalized Hadamard codes ✩,✩✩ Adrián Torres-Martín ∗ , Mercè Villanueva Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain a r t i c l e i n f o a b s t r a c t Article history: It is known that Zps -linear codes, which are the Gray map Received 6 March 2023 image of Zps -additive codes (linear codes over Zps ), are Received in revised form 20 systematic and a systematic encoding has been found. This September 2023 makes Zps -linear codes suitable to apply the permutation Accepted 5 October 2023 Available online 20 October 2023 decoding method. This technique is also based on the Communicated by Gary L. Mullen existence of r-PD-sets, which are subsets of the permutation automorphism group of the code. In this paper, we study the MSC: permutation automorphism group of Zps -linear generalized 94B25 Hadamard codes of type (n; t1 , . . . , ts ) and show how to 94B60 construct r-PD-sets of size r + 1, for all r up to an upper bound, in order to be able to perform a partial permutation Keywords: decoding for these codes. Permutation decoding © 2023 The Author(s). Published by Elsevier Inc. This is an PD-set open access article under the CC BY-NC-ND license Automorphism group Generalized Hadamard code (http://creativecommons.org/licenses/by-nc-nd/4.0/). Zps -linear code Gray map ✩ This work has been partially supported by the Spanish Ministerio de Ciencia e Innovación under Grants PID2019-104664GB-I00, PID2022-137924NB-I00, and RED2022-134306-T (AEI / 10.13039/501100011033) and by the Agència de Gestió d’Ajuts Universitaris i de Recerca, Generalitat de Catalunya grant 2021SGR 00643. ✩✩ The material in this paper was presented in part at the 2022 IEEE Information Theory Workshop in Mumbai, India, November 2022 [33]. * Corresponding author. E-mail address: adrian.torres@uab.cat (A. Torres-Martín). https://doi.org/10.1016/j.ffa.2023.102316 1071-5797/© 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 1. Introduction Let Zps be the ring of integers modulo ps with s ≥ 1 and p prime, and Znps be the set of n-tuples over Zps . In this paper, the elements of Znps are also called vectors over Zps of length n. A code over Zp of length n is a nonempty subset of Znp , and it is linear if it is a subspace of Znp . A nonempty subset of Znps is a Zps -additive code if it is a subgroup of Znps . Note that, when p = 2 and s = 1, a Zps -additive code is a binary linear code and, when p = 2 and s = 2, it is a quaternary linear code or a linear code over Z4 . Let Sn be the symmetric group of permutations on the set {1, . . . , n}. Two codes over Zp of length n, C1 and C2 , are said to be permutation equivalent if there is a permutation of coordinates π ∈ Sn such that C2 = {π(c) : c ∈ C1 }. Two Zps -additive codes of length n, C1 and C2 , are said to be permutation equivalent if they differ only by a permutation of coordinates, that is, if there is a permutation of coordinates π ∈ Sn such that C2 = {π(c) : c ∈ C1 }. The Hamming weight of a vector u ∈ Znp , denoted by wtH (u), is the number of nonzero coordinates of u. The Hamming distance of two vectors u, v ∈ Znp , denoted by dH (u, v), is the number of coordinates in which they differ. Note that dH (u, v) = wtH (v − u). The minimum distance of a code C over Zp is d(C) = min{dH (u, v) : u, v ∈ C, u = v}. For elements of Zps , we consider the following metric, defined in [12], and also used in [19,31]: ⎧ ⎪ ⎪ ⎨0 if x = 0, ∗ wt (x) = p s−1 if x ∈ ps−1 Zps \{0}, (1) ⎪ ⎪ ⎩(p − 1)ps−2 otherwise. n The weight of a vector u = (u1 , u2 , . . . , un ) ∈ Znps is wt∗ (u) = j=1 wt∗ (uj ) ∈ Zps ; and the distance between two vectors u, v ∈ Znps is d∗ (u, v) = wt∗ (u − v). The minimum distance of a code C over Zps is d∗ (C) = min{d∗ (u, v) : u, v ∈ C, u = v}. In [20,27], a Gray map from Z4 to Z22 is defined as φ(0) = (0, 0), φ(1) = (0, 1), φ(2) = (1, 1) and φ(3) = (1, 0). There exist different generalizations of this Gray map, s−1 which go from Z2s to Z22 [10,13,23]. The one given by Krotov in [23] is defined in terms of the codewords of a Hadamard code, and the one given by Carlet in [10] is a   particular case of Krotov’s one satisfying λi φs (2i ) = φs ( λi 2i ) [15]. In this paper, we consider a generalization of Carlet’s Gray map, denoted by φs and defined as follows: φs (u) = (us−1 , . . . , us−1 ) + (u0 , . . . , us−2 )Ys−1 , (2) s−1 where u ∈ Zps , [u0 , u1 , . . . , us−1 ]p is the p-ary expansion of u, that is u = i=0 pi ui (ui ∈ Zp ), and Ys−1 is a matrix of size (s − 1) × ps−1 whose columns are the elements of Zs−1 p . Note that the rows of Ys−1 form a basis of a first order Reed-Muller code after adding the all-one row. This Gray map φs is an isometric embedding from (Zps , d∗ ) s−1 into (Zpp , dH ) [19,31]. If s = 1, then φs is the identity map. In order to simplify the A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 3 notation, we write φ instead of φs , when s is clear from the context. Then, we define s−1 Φ : Znps → Znp p as the component-wise extension of φ. Let C be a Zps -additive code of length n. We say that its Gray map image, C = Φ(C), is a Zps -linear code of length ps−1 n. Since C is a subgroup of Znps , it is isomorphic to an abelian structure Ztp1s × Ztp2s−1 × · · · × Ztps , and we say that C, or equivalently C = Φ(C), is of type (n; t1 , . . . , ts ). Note that |C| = pst1 p(s−1)t2 · · · pts . Unlike linear codes over finite fields, linear codes over rings do not have a basis, but there exists a generator matrix for these codes having minimum number of rows, that is, t1 + · · · + ts rows. A generalized Hadamard (GH) matrix H(p, λ) = (hij ) of order N = pλ over Zp is a pλ × pλ matrix with entries in Zp with the property that, for every i, j, 1 ≤ i < j ≤ pλ, each of the multisets {hik − hjk : 1 ≤ k ≤ pλ} contains every element of Zp exactly λ times [22]. Two GH matrices H1 and H2 of order N are said to be equivalent if one can be obtained from the other by a permutation of the rows and columns and adding the same element of Zp to all the coordinates in a row or in a column. We can always change the first row and column of a GH matrix into zeros, obtaining an equivalent GH matrix which is called normalized. From a GH matrix H, the generalized Hadamard (GH) code  is CH = α∈Zp (FH +α1), where FH +α1 = {h+α1 : h ∈ FH }, FH is the code consisting of the rows of H, and 1 denotes the all-one vector [14]. Note that CH is not necessarily linear as a code over Zp . A Zps -additive code C such that Φ(C) is a GH code is called a Zps -additive GH code and Φ(C) is called a Zps -linear GH code. Note that a GH code over Zp of length N has pN codewords and minimum distance (p − 1)N/p. The Z4 -linear Hadamard codes of length 2t have been studied and classified in [24,29], and their automorphism groups have been characterized in [25,28]. For s > 2, Z2s -linear Hadamard codes where first introduced in [23]. A full classification of Z8 -linear Hadamard codes is provided in [16]. For s > 3, a partial classification and bounds on the number of nonequivalent Z2s -linear Hadamard codes of length 2t can be found in [15]. More generally, for any s ≥ 2 and p prime, Zps -linear GH codes are studied and partially classified in [5,6]. Moreover, it is proved that, for p ≥ 3, the Zps -linear GH codes of type (n; 1, 0, . . . , 0, ts ) are the only ones which are linear [5]. For p = 2, they are only linear when their type is (n; 1, 0, . . . , 0, ts ) or (n; 1, 0, . . . , 0, 1, ts ) [15]. Let C be a code over Zp of length n with pk codewords. For a vector u ∈ Znp and a set I ⊆ {1, . . . , n}, we denote the projection of u to the coordinates of I by u|I . We say that C is a systematic code if there is a set I ⊆ {1, . . . , n} of k coordinate positions such that |CI | = pk , where CI = {u|I : u ∈ C}. The set I is called an information set for C and {1, . . . , n}\I a redundancy set. Permutation decoding is a technique, introduced by Prange [30] and developed by MacWilliams [26] for linear codes, that involves finding a subset of the permutation au- tomorphism group of a code in order to assist in decoding. In [4], a new permutation decoding method for Z4 -linear codes (not necessarily linear), based on having a sys- tematic encoding for these codes, was introduced. Actually, it is also proved that this method can be used for any nonlinear binary code, as long as it has a systematic en- 4 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 coding. This can be generalized easily to systematic nonlinear codes over Zp [32]. Then, since any Zps -linear code is systematic, as shown in [32] by giving a systematic encoding, the permutation decoding method can also be used for these codes. The idea behind the permutation decoding technique is to move all errors in a received vector out of the information positions by using a permutation that preserves the code. Let C be a t-error-correcting code over Zp and denote by PAut(C) its permutation automorphism group. Then, it is necessary to find a subset S ⊆ PAut(C), with respect to an information set for C, such that every r-set of coordinate positions is moved out of the information coordinates by at least one element in S, where 1 ≤ r ≤ t. The set S is called an r-PD-set and, if r = t, it is called a PD-set. The efficiency of the permutation decoding method depends on the size of the r-PD- set S ⊆ PAut(C), since it needs to find the suitable permutation in S, for each received vector. In general, determining the structure of PAut(C) is very complex, making the search for r-PD-sets or PD-sets a difficult task. However, there are results that show how to find r-PD-sets of small size for certain families of codes [2,3,11,18]. More specifically, in [2], it is shown how to find r-PD-sets of size r + 1 for binary linear Hadamard codes and (nonlinear) Z4 -linear Hadamard codes. A similar result for Hadamard codes over the field F4 is presented in [11]. In this paper, we generalize these results to Zps -linear GH codes with s ≥ 2 and p prime. The paper is organized as follows. In Section 2, we recall the recursive construction of Zps -additive GH codes. In Section 3, we study the permutation automorphism group for these codes and show that it is isomorphic to a group formed by matrices of the general linear group over Zps . In Section 4, we give an information set for the corresponding Zps - linear GH codes and establish a criterion to find r-PD-sets of size r + 1. In Sections 5 and 6, explicit and recursive constructions of r-PD-sets of this size, up to a given upper bound, are described. In Section 7, we present some computational results on a random search of these sets for the codes where the upper bound is not reached. Finally, in Section 8, some conclusions and further research on this topic are included. 2. Construction of Zps -additive GH codes Generator matrices having minimum number of rows for Z4-additive Hadamard codes, as well as a recursive construction of these matrices, are given in [24]. In [23], Z2s -additive Hadamard codes with s > 2 are introduced and generator matrices with minimum num- ber of rows are also given. A recursive construction for these matrices is presented in [15]. More recently, Zps -additive GH codes are considered in [5] generalizing these results to any p ≥ 3 prime. In this section, we recall the construction of Zps -additive GH codes with s ≥ 2 and p prime. Let t1 , t2 ,. . . ,ts be nonnegative integers with t1 ≥ 1. Consider the matrix G t1 ,...,ts whose columns are exactly all the vectors of the form zT , z ∈ {1} × Ztp1s−1 × (pZps ) 2 × t  s−1 ts · · · × p Zps . A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 5 Example 2.1. For p = 3 and s = 3, we have the following matrices over Z27 : 11 1 1111 1 1 1 1 1 G 1,0,1 = , G 1,1,0 = , 0 9 18 0 3 6 9 12 15 18 21 24 1111111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 G 2,0,0 = , 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 ⎛ ⎞ 1111 1 1 1 1 1 1111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎜ ⎟ G 1,1,1 = ⎝ 0 3 6 9 12 15 18 21 24 0 3 6 9 12 15 18 21 24 0 3 6 9 12 15 18 21 24 ⎠ , 0 0 0 0 0 0 0 0 0 9 9 9 9 9 9 9 9 9 18 18 18 18 18 18 18 18 18 ⎛ ⎞ 1 1 1 ··· 1 1 1 1 ··· 1 1 1 1 ··· 1 ⎜ ⎟ G 2,0,1 = ⎝ 0 1 2 · · · 26 0 1 2 · · · 26 0 1 2 · · · 26 ⎠ . 0 0 0 · · · 0 9 9 9 · · · 9 18 18 18 · · · 18 Let 0, 1, 2, . . . , ps − 1 be the vectors having the same element 0, 1, 2, . . . , ps − 1 from Zps in all its coordinates, respectively. The order of a vector u over Zps , denoted by ord(u), is the smallest positive integer m such that mu = 0. Any matrix G t1 ,...,ts can also be obtained by applying the following recursive con- struction. We start with G 1,0,...,0 = (1). Then, if we have a matrix G = G t1 ,...,ts , for any i ∈ {1, . . . , s}, we may construct the matrix G G ··· G Gi = . (3) 0 · pi−1 1 · pi−1 ··· (ps−i+1 − 1) · pi−1   Finally, permuting the rows of Gi , we obtain a matrix G t1 ,...,ts , where tj = tj for j = i   and ti = ti + 1. Note that any permutation of columns of Gi gives also a matrix G t1 ,...,ts . Example 2.2. From the matrix G 1,0,0 = (1), we obtain the matrix G 2,0,0 ; and from G 2,0,0 we can construct G 2,0,1 , where G 2,0,0 and G 2,0,1 are the matrices given in Example 2.1. Note that we can also generate another matrix G 2,0,1 as follows: from G 1,0,0 = (1) we obtain the matrix G 1,0,1 given in Example 2.1, and from G 1,0,1 we can construct the matrix ⎛ ⎞ 1 1 1 1 1 1 1 1 1 ··· 1 1 1 1 1 1 ⎜ ⎟ G1 = ⎝ 0 9 18 0 9 18 0 9 18 · · · 0 9 18 0 9 18 ⎠ . 0 0 0 1 1 1 2 2 2 · · · 25 25 25 26 26 26 Then, after permuting the rows of G1 , we have the matrix ⎛ ⎞ 1 1 1 1 1 1 1 1 1 ··· 1 1 1 1 1 1 ⎜ ⎟ G 2,0,1 = ⎝ 0 0 0 1 1 1 2 2 2 · · · 25 25 25 26 26 26 ⎠ , 0 9 18 0 9 18 0 9 18 · · · 0 9 18 0 9 18 6 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 which is different to the matrix G 2,0,1 given in Example 2.1. Note that these two matrices G 2,0,1 generate permutation equivalent codes. In this paper, we assume that the matrices G t1 ,...,ts are constructed recursively starting from G 1,0,...,0 in the following way. First, we obtain G t1 ,0,...,0 by adding t1 − 1 rows of order ps ; then G t1 ,t2 ,0,...,0 is generated by adding t2 rows of order ps−1 ; and so on, until G t1 ,...,ts is reached by adding ts rows of order p. We denote by Ht1 ,...,ts the Zps -additive code of type (n; t1 , . . . , ts ) generated by G t1 ,...,ts , where t1 , . . . , ts are nonnegative integers with t1 ≥ 1. Note that n = pt−s+1 , s where t = ( i=1 (s − i + 1) · ti ) − 1. Let H t1 ,...,ts = Φ(Ht1 ,...,ts ) denote the correspond- ing Zps -linear code, which is a GH code of length pt [5]. Thus, we say that Ht1 ,...,ts is a Zps -additive GH code, and H t1 ,...,ts a Zps -linear GH code. 3. Permutation automorphism group of Zps -additive GH codes The structure of the permutation automorphism group of Z4 -additive Hadamard codes is described in [25,28]. In this section, we describe the structure of the permu- tation automorphism group of the Zps -additive GH code Ht1 ,...,ts , that is, the structure of PAut(Ht1 ,...,ts ). In particular, an isomorphism between PAut(Ht1 ,...,ts ) and a cer- tain group of matrices of the general linear group over Zps is found, and the order of PAut(Ht1 ,...,ts ) is computed. Let GL(κ, Zps ) denote the general linear group of degree κ over Zps and let L be the set consisting of all matrices over Zps of the following form: ⎛ ⎞ 1 a1 pa2 p 2 a3 ··· ps−2 as−1 ps−1 as ⎜0 A1,1 pA1,2 p2 A1,3 ··· ps−2 A1,s−1 ps−1 A1,s ⎟ ⎜ ⎟ ⎜0 ··· ps−3 A2,s−1 ps−2 A2,s ⎟ ⎜ A2,1 A2,2 pA2,3 ⎟ ⎜ ⎟ ⎜0 A3,1 A3,2 A3,3 ··· ps−4 A3,s−1 ps−3 A3,s ⎟ , (4) ⎜. ⎟ ⎜. .. .. .. .. .. .. ⎟ ⎜. . . . . . . ⎟ ⎜ ⎟ ⎝0 As−1,1 As−1,2 As−1,3 ··· As−1,s−1 pAs−1,s ⎠ 0 As,1 As,2 As,3 ··· As,s−1 As,s where A1,1 ∈ GL(t1 − 1, Zps ), Ai,i ∈ GL(ti , Zps ) for i ∈ {2, . . . , s}, Ai,j are matrices over Zps , for i = j, a1 ∈ Ztp1s−1 and aj ∈ Zpjs , for j ∈ {2, . . . , s}. t Lemma 3.1. The set L is a subgroup of GL(t1 + · · · + ts , Zps ). Proof. We first need to check that L ⊆ GL(t1 + · · · + ts , Zps ), i.e., that det(M) ∈ Zps \pZps for all M ∈ L. Note that if M ∈ GL(κ, Zps ), then M = M +pR ∈ GL(κ, Zps ) for any R. Thus, since det(M ) ∈ Zps \pZps , we have that det(M) ∈ Zps \pZps , where A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 7 ⎛ ⎞ 1 a1 0 0 ··· 0 0 ⎜0 A1,1 0 0 ··· 0 0 ⎟ ⎜ ⎟ ⎜0 ··· 0 ⎟ ⎜ A2,1 A2,2 0 0 ⎟ ⎜ ⎟  ⎜ M =⎜ 0 A3,1 A3,2 A3,3 ··· 0 0 ⎟. ⎜ .. .. .. .. .. .. .. ⎟⎟ ⎜. . . . . . . ⎟ ⎜ ⎟ ⎝ 0 As−1,1 As−1,2 As−1,3 ··· As−1,s−1 0 ⎠ 0 As,1 As,2 As,3 ··· As,s−1 As,s Finally, we prove that L is a subgroup. Let us denote by Mi,j , for i, j ∈ {1, . . . , s + 1}, the submatrix in the ith row and jth column of the block matrix M ∈ L as given in (4). Note that M1,j is a multiple of pj−2 for j ∈ {2, . . . , s + 1}, and Mi,j is a multiple of pj−i for i, j ∈ {2, . . . , s + 1} and j > i. Then, consider the submatrix Qi,j of Q = MN , for M, N ∈ L. Clearly, Q1,1 = 1 and Qi,1 = 0 for i ∈ {2, . . . , s + 1}. For the first row, s+1 we have Q1,j = k=1 M1,k Nk,j for j ∈ {2, . . . , s + 1}. Note that M1,1 N1,j = N1,j is a multiple of pj−2 , M1,k Nk,j is a multiple of pk−2 pj−k = pj−2 for k ∈ {2, . . . , j}, and a multiple of pk−2 for k ∈ {j + 1, . . . , s + 1}. Therefore, Qi,j is a multiple of pj−2 . For the s+1 rest of the rows, Qi,j = k=2 Mi,k Nk,j for i, j ∈ {2, . . . , s + 1} and j > i. Note that Mi,k Nk,j is a multiple of pj−k for k ∈ {2, . . . , i − 1}, a multiple of pk−i pj−k = pj−i for k ∈ {i, . . . , j}, and a multiple of pk−i for k ∈ {j + 1, . . . , s + 1}. Therefore, Qi,j is also a s+1 multiple of pj−i . Finally, the block submatrices in the diagonal are Qi,i = k=2 Mi,k Nk,i for i ∈ {2, . . . , s + 1}. Note that Mi,i Ni,i ∈ GL(ti , Zps ) and Mi,k Nk,i is a multiple of pi−k for k < i and a multiple of pk−i for k > i, hence Qi,i ∈ GL(ti , Zps ).  Let ζi be the map from Zps to Zps defined as ζi (a) = a mod pi , i ∈ {1, . . . , s − 1}. This map can be extended to matrices over Zps by applying ζi to each one of their entries. Let π be the map from L to L defined as ⎛ ⎞ 1 a1 pa2 ··· ps−2 as−1 ps−1 as ⎜0 ··· ps−2 A1,s−1 ps−1 A1,s ⎟ ⎜ A1,1 pA1,2 ⎟ ⎜ ⎟ ⎜ 0 ζs−1 (A2,1 ) ζs−1 (A2,2 ) · · · ζs−1 (p A2,s−1 ) ζs−1 (p A2,s ) ⎟ s−3 s−2 π(M) = ⎜ ⎜ .. .. .. .. .. .. ⎟, ⎟ (5) ⎜ . . . . . . ⎟ ⎜ ⎟ ⎝ 0 ζ2 (As−1,1 ) ζ2 (As−1,2 ) ··· ζ2 (As−1,s−1 ) ζ2 (pAs−1,s ) ⎠ 0 ζ1 (As,1 ) ζ1 (As,2 ) ··· ζ1 (As,s−1 ) ζ1 (As,s ) for any matrix M ∈ L as given in (4). Let π(L) = {π(M) : M ∈ L} ⊆ GL(t1 + · · · + ts , Zps ). By Lemma 3.1, it is clear that π(L) is a group with the operation ∗ defined as M ∗ N = π(MN ) for all M, N ∈ π(L). Note that the group operation ∗ is well defined, since π(L) ⊆ L. By a generalization of the proof of Theorem 2 in [25], one can show the following theorem. Theorem 3.1. Let Ht1 ,...,ts be the Zps -additive GH code of type (n; t1 , . . . , ts ). Then, PAut(Ht1 ,...,ts ) is isomorphic to π(L). 8 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 Proof. Let R be the set Ztp1s−1 × Ztp2s−1 × · · · × Ztps and denote by B the set of all affine functions from R to Zps . We can see these Zps -valued affine functions as words of length n = ps(t1 −1)+(s−1)t2 +···+ts over Zps by considering the image of all elements in the s−1 domain. Let us define B = {x : R −→ Zpp | x(·) = φ(f (·)) for some f ∈ B}. That is, the image of the words in B by the generalized Gray map Φ. By a straightforward generalization of Lemma 1 in [25], we know that B is a Zps -linear GH code of type (n; t1 , t2 , . . . , ts ). This means that Ht1 ,...,ts = B and H t1 ,...,ts = B, and we can see the elements of Ht1 ,...,ts as affine functions. Note that an affine function f ∈ B can be seen as a word wf of length n over Zps , with the elements of R playing the role of coordinate positions. A permutation σ ∈ Sn acting on wf , by permuting its coordinates, gives a word σ(wf ) which corresponds to the function f ◦ σ −1 by considering f (σ −1 (v)) for any v ∈ R. Therefore, a permutation σ is said to be in PAut(B) if and only if f ◦ σ −1 is an affine function for any f ∈ B. Naturally, PAut(B) = PAut(Ht1 ,...,ts ). Now, we use an adaptation of Theorem 2 in [25] to prove that PAut(Ht1 ,...,ts ), or equivalently PAut(B), consists of all affine permutations of R. First, we have that any affine permutation belongs to PAut(B) since the composition of an affine permutation and an affine function is also an affine function. Next, we see that any permutation σ ∈ PAut(B) is affine. Let (1) (1) σ1 , . . . , σt1 −1 : R −→ Zps , (2) (2) σ1 , . . . , σt2 : R −→ Zps−1 , .. . (s) (s) σ1 , . . . , σts : R −→ Zp , be the components of σ −1 . That is, σ −1 (v) = (σ1 (v), . . . , σt1 −1 (v), σ1 (v), . . . , σt2 (v), . . . , σ1 (v), . . . , σts (v)) (1) (1) (2) (2) (s) (s) for any v ∈ R. Consider the following functions defined from R to Zps : (j) (1) (1) (2) (2) (s) (s) (j) fi (x1 , . . . , xt1 −1 ,x1 , . . . , xt2 , . . . , x1 , . . . , xts ) = pj−1 xi ,  for i ∈ {1, . . . , t1 − 1} if j = 1, for i ∈ {1, . . . , tj } if j ∈ {2, . . . , s}. (j) (j) Note that pj−1 xi defines the inclusion of xi ∈ Zps−j+1 in Zps . These functions are affine, hence fi ∈ B and, since σ ∈ PAut(B), we have that fi ◦ σ −1 ∈ B. Moreover, (j) (j) fi (σ (v)) = pj−1 σi (v), therefore σi is affine. Since all components are affine, σ −1 (j) −1 (j) (j) and σ are also affine. A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 9 Finally, we show that the group of affine permutations over R is isomorphic to π(L). Let us denote the former by S. Then, we see that S is isomorphic to π(L) via the map ψ, defined from π(L) to S as ψ(π(M)) = σ(u1 , u2 , . . . , us ) = b + u1 A1 + u2 pA2 + · · · + us ps−1 As , where b = (a1 , pa2 , . . . , ps−1 as ) and A1 = (A1,1 , pA1,2 , . . . , ps−1 A1,s ), A2 = (ζs−1 (A2,1 ), ζs−1 (A2,2 ), . . . , ζs−1 (ps−2 A2,s )), .. . As = (ζ1 (As,1 ), ζ1 (As,2 ), . . . , ζ1 (As,s )). Note that A1 , pA2 , . . . , ps−1 As are matrices over Zps with linearly independent rows of order ps , ps−1 , . . . , p, respectively, spanning R. This is ensured due to Aj,j , for j ∈ {1, . . . , s}, being invertible. The map ψ gives an isomorphism between π(L) and S, so π(L) and PAut(Ht1 ,...,ts ) are isomorphic.  Theorem 3.2. Let Ht1 ,...,ts be the Zps -additive GH code of type (n; t1 , . . . , ts ), where n = ps(t1 −1)+(s−1)t2 +···+ts . Let t̄1 = t1 − 1 and t̄i = ti for i ∈ {2, . . . , s}. The order of its permutation automorphism group is | PAut(Ht1 ,...,ts )| = pE N1 · · · Ns , (6) 2 t̄i (t̄i −1) t̄i where Ni = |GL(t̄i , Zps−i+1 )| = p(s−i)t̄i + 2 j=1 (p j − 1) and  s  s E = st̄1 + (s − 1)t̄2 + · · · + t̄s + 2(s − j + 1)t̄i t̄j . (7) i=1 j=i+1 Proof. The order of PAut(Ht1 ,...,ts ) can be easily computed with a counting argument over the matrix representation, that is, over the elements of π(L), given in (5). The first row of a matrix M ∈ π(L) is a random tuple over Zpt1s−1 × Ztp2s−1 × · · · × Ztps . There are ps(t1 −1)+(s−1)t2 +···+ts such tuples. Note that ζs−i+1 (pj−i Ai,j ), for i < j, is a matrix of size t̄i × t̄j defined over Zps , with entries among ps−i+1 /pj−i = ps−j+1 elements in Zps . In the case i = 1, ζs represents the identity map from Zps to Zps . Therefore, there are p(s−j+1)t̄i t̄j such matrices. Moreover, ζs−j+1 (Aj,i ), for i < j, is a matrix of size t̄j × t̄i with entries among the same number of elements as ζs−i+1 (pj−i Ai,j ). Then, for each pair i, j ∈ {1, . . . , s} such that i < j, we have p2(s−j+1)t̄i t̄j different possibilities to choose the corresponding matrices. All matrices in the diagonal are invertible matrices defined over Zps . Moreover, ζs−i+1 (Ai,i ) can be represented as an invertible matrix over Zps−i+1 by considering 10 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 {0, . . . , ps−i+1 − 1} ⊂ Zps as elements in Zps−i+1 . Therefore, ζs−i+1 (Ai,i ) is a matrix in the group GL(t̄i , Zps−i+1 ). In [21], the order of the general linear group over inte- gers modulo m is given. In particular, the order of GL(κ, Zps ), denoted by νκ (ps ), for 2 p prime and integers κ and s ≥ 2, satisfies νκ (ps ) = p(s−1)κ νκ (p), where νκ (p) = (pκ − 1)(pκ − p) · · · (pκ − pκ−1 ) is the order of the general linear group over the field Zp. 2 t̄i (t̄i −1) t̄ Then, the order of GL(t̄i , Zps−i+1 ) is νt̄i (Zps−i+1 ) = p(s−i)t̄i + 2 j=1 (p − 1). i j Considering all possible choices of submatrices, the result follows.  Remark 3.1. If we consider the case with p = 2 and s = 2, that is, Z4 -additive Hadamard codes of type (n; t1 , t2 ), then (7) becomes E = 2(t1 − 1) + t2 + 2(t1 − 1)t2 . We also have (t1 −1)(t1 −2) t −1 N1 = |GL(t1 − 1, Z4 )| = 2(t1 −1) + 2 j=1 (2 − 1) and N2 = |GL(t2 , Z2 )| = 2 1 j t2 (t2 −1) t j=1 (2 − 1). Therefore, 2 j 2 2 1 −1 t  t2 3(t1 −1)2 3(t1 −1) t2 t2 +2(t1 −1)t2 + 22 | PAut(Ht1 ,t2 )| = 2 2 + 2 + 2 (2j − 1) (2j − 1). j=1 j=1 Note that this expression coincides with the one given in [2]. Example 3.1. Consider the Z27 -additive GH code H2,1,1 . By Theorem 3.1, PAut(H2,1,1 ) is isomorphic to the group π(L) ⊆ GL(4, Z27 ). The subgroup L ⊆ GL(4, Z27 ) is formed by all matrices in the form ⎛ ⎞ 1 a1 3a2 9a3 ⎜0 A 9A1,3 ⎟ ⎜ 1,1 3A1,2 ⎟ ⎜ ⎟, ⎝ 0 A2,1 A2,2 3A2,3 ⎠ 0 A3,1 A3,2 A3,3 where a1 , a2 , a3 , Ai,j ∈ Z27 , i, j ∈ {1, 2, 3} with i = j, and A1,1 , A2,2 , A3,3 ∈ Z27 \ {0, 3, 6, 9, 12, 15, 18, 21, 24}. For example, consider the following two matrices M, N ∈ L: ⎛ ⎞ ⎛ ⎞ 1 1 21 9 1 19 18 0 ⎜ 0 2 3 18 ⎟ ⎜0 8 9⎟ ⎜ ⎟ ⎜ 24 ⎟ M=⎜ ⎟, N =⎜ ⎟. ⎝ 0 14 14 0 ⎠ ⎝ 0 18 20 0⎠ 0 9 16 22 0 16 4 7 The function π reduces the third row modulo 9 and the fourth row modulo 3, therefore ⎛ ⎞ ⎛ ⎞ 1 1 21 9 1 19 18 0 ⎜0 18 ⎟ ⎜0 8 24 9 ⎟ ⎜ 2 3 ⎟ ⎜ ⎟ π(M) = ⎜ ⎟, π(N ) = ⎜ ⎟. ⎝0 5 5 0 ⎠ ⎝0 0 2 0⎠ 0 0 1 1 0 1 1 1 Note that, since L is a group, π(L) is also a group with the operation ∗: A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 11 ⎛⎛ ⎞⎞ ⎛ ⎞ 1 9 12 18 1 9 12 18 ⎜⎜ ⎟⎟ ⎜ ⎟ ⎜⎜ 0 7 18 9 ⎟⎟ ⎜ 0 7 18 9 ⎟ π(M) ∗ π(N ) = π(π(M)π(N )) = π ⎜⎜ ⎟⎟ = ⎜ ⎟. ⎝⎝ 0 13 22 18 ⎠⎠ ⎝ 0 4 4 0 ⎠ 0 1 3 1 0 1 0 1 Theorem 3.2 gives the order of π(L), or equivalently of PAut(H2,1,1 ). Following the same notation as in the statement of the theorem, we have that N1 = |GL(1, Z27 )| = 18, N2 = |GL(1, Z9 )| = 6, N3 = |GL(1, Z3 )| = 2, E = 3 + 2 + 1 + 4 + 2 + 2 = 14. Therefore, | PAut(H2,1,1 )| = 3E N1 N2 N3 = 317 · 23 = 1033121304. 4. r-PD-sets for Zps -linear GH codes In this section, we give an additive information set for the Zps -additive GH code Ht1 ,...,ts , and an information set for the corresponding Zps -linear GH code H t1 ,...,ts . Then, using the result given by Theorem 3.1, we establish a criterion in order to find r-PD-sets of size r + 1 for H t1 ,...,ts , with r up to a given upper bound. An ordered set I = {i1 , . . . , it1 +···+ts } ⊆ {1, . . . , n} of t1 + · · · + ts coordinate positions is said to be an additive information set for a Zps -additive code C of type (n; t1 , . . . , ts ) if |CI | = (ps )t1 (ps−1 )t2 · · · pts . If the elements of I are ordered in such a way that, for any k ∈ {1, . . . , s}, |C{i1 ,...,it1 +···+tk } | = (ps )t1 (ps−1 )t2 · · · (ps−k+1 )tk , then it can be seen that the set Φ(I), defined as Φ(I) =Φ(1) ({i1 , . . . , it1 }) ∪ Φ(2) ({it1 +1 , . . . , it1 +t2 })∪ · · · ∪ Φ(s) ({it1 +···+ts−1 +1 , . . . , it1 +···+ts }), where Φ(k) (I) = ∪i∈I {ps−1 (i − 1) + 1, ps−1 (i − 1) + pk−1 + 1, ps−1 (i − 1) + pk−1+1 + 1, ps−1 (i − 1) + pk−1+2 + 1, ..., ps−1 (i − 1) + ps−2 + 1}, is an information set for C = Φ(C) [32]. Note that s − 2 − (k − 1) = s − k − 1, hence Φ(k) (I) has s − k + 1 coordinate positions for each element in I. 12 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 Example 4.1. It is easy to see, from the matrix G 1,1,1 given in Example 2.1, that the set I = {1, 2, 10} is an additive information set for the Z27 -additive GH code H1,1,1 , so Φ(I) = Φ(1) ({1}) ∪ Φ(2) ({2}) ∪ Φ(3) ({10}) = {1, 2, 4, 10, 13, 82} is an information set for H 1,1,1 = Φ(H1,1,1 ). In general, there is no unique way to obtain an additive information set for Ht1 ,...,ts . The following result provides a recursive and simple form to obtain such a set. Proposition 4.1. Let I be an additive information set for the Zps -additive GH code Ht1 ,...,ts of type (n; t1 , . . . , ts ), where n = ps(t1 −1)+(s−1)t2 +···+ts . Then I ∪ {n + 1} is an additive information set for the codes Ht1 +1,t2 ,...,ts , Ht1 ,t2 +1,...,ts and so on until Ht1 ,t2 ,...,ts +1 , obtained from Ht1 ,t2 ,...,ts by applying (3).    Proof. Let Hk = Ht1 ,t2 ,...,ts , k ∈ {1, . . . , s}, where tj = tj for j = k and tk = tk + 1. It is clear that an additive information set for Hk should have t1 + t2 + · · · + ts + 1 = |I| + 1 coordinate positions. Taking into account that Hk is constructed from Ht1 ,t2 ,...,ts by applying (3), we have that |(Hk )I∪{x} | = (ps )t1 (ps−1 )t2 · · · pts ps+1−k for all x ∈ {n + 1, . . . , 2n}. In particular, I ∪ {n + 1} is an additive information set for Hk .  Let I be an additive information set for Ht1 ,...,ts of type (n; t1 , . . . , ts ). Let Hk = t1 ,t2 ,...,ts H , k ∈ {1, . . . , s}, where tj = tj for j = k and tk = tk + 1. Although the additive information set I ∪ {n + 1}, given by Proposition 4.1, is the same for all Hk , the information sets for the corresponding Zps -linear codes over Zp , Hk = Φ(Hk ), differ for every k ∈ {1, . . . , s}. In particular, I (k) = Φ(I) ∪ {ps−1 n + 1, ps−1 n + pk−1 + 1, pk−1 n + pk + 1, . . . , ps−1 n + ps−2 + 1} is an information set for Hk . We can label the ith coordinate position of a Zps -additive GH code Ht1 ,...,ts , with the ith column of its generator matrix G t1 ,...,ts . Note that, by construction, all columns in G t1 ,...,ts are different and there are n = ps(t1 −1)+(s−1)t2 +···+ts of them. Thus, any additive information set I for Ht1 ,...,ts can also be considered as a set of vectors representing the positions in I. Let ei be the vector with all coordinates equal to 0 except the one in the ith position, which is equal to 1. Then, by Proposition 4.1, we have that the set It1 ,...,ts ={e1 , e1 + e2 , . . . , e1 + et1 } ∪ {e1 + pet1 +1 , . . . , e1 + pet1 +t2 } ∪ · · · ∪ {e1 + ps−1 et1 +t2 +···+ts−1 +1 , . . . , e1 + ps−1 et1 +t2 +···+ts } is a suitable additive information set for Ht1 ,...,ts . Depending on the context, It1 ,...,ts is considered as a subset of {1, . . . , n} or {1} × Ztp1s−1 × (pZps )t2 × · · · × (ps−1 Zps )ts . A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 13 Example 4.2. Let H2,0,0 be the Z27 -additive GH code of length 27 with generator matrix G 2,0,0 given in Example 2.1. The set I2,0,0 = {1, 2}, or equivalently the set of vectors I2,0,0 = {e1 , e1 + e2 }, is an additive information set for H2,0,0 . By applying (3) over G 2,0,0 , we obtain matrices G 3,0,0 , G 2,1,0 and G 2,0,1 that gen- erate the Z27 -additive GH codes H3,0,0 , H2,1,0 and H2,0,1 of length 729, 243 and 81, respectively. By Proposition 4.1, it follows that I2,0,0 ∪ {28} = {1, 2, 28} is an additive information set for H3,0,0 , H2,1,0 and H2,0,1 . Although this additive information set is the same for these three codes, it is important to note that in terms of vectors representing these positions, we have that I3,0,0 = {(1, 0, 0), (1, 1, 0), (1, 0, 1)}, I2,1,0 = {(1, 0, 0), (1, 1, 0), (1, 0, 3)}, and I2,0,1 = {(1, 0, 0), (1, 1, 0), (1, 0, 9)}. Finally, I (1) = Φ(I2,0,0 ) ∪ {244, 245, 247} = {1, 2, 4, 10, 11, 13, 244, 245, 247}, I (2) = Φ(I2,0,0 ) ∪ {244, 247} = {1, 2, 4, 10, 11, 13, 244, 247}, and I (3) = Φ(I2,0,0 ) ∪ {244} = {1, 2, 4, 10, 11, 13, 244} are information sets for the corresponding Z27 -linear GH codes H 3,0,0 , H 2,1,0 and H 2,0,1 , respectively. Let C be a Zps -additive code of type (n; t1 , . . . , ts ), and let C = Φ(C) be the corre- sponding Zps -linear code of length ps−1 n. Let Φ : Sym(n) → Sym(ps−1 n) be the map defined as   i + χ(i) Φ(τ )(i) = ps−1 τ − χ(i), (8) ps−1 where χ(i) = ps−1 − (i mod ps−1 ), for all τ ∈ Sym(n) and i ∈ {1, . . . , ps−1 n}. Given a subset S ⊆ Sym(n), we define the set Φ(S) = {Φ(τ ) : τ ∈ S} ⊆ Sym(ps−1 n). It is easy to see that if S ⊆ PAut(C) ⊆ Sym(n), then Φ(S) ⊆ PAut(Φ(C)) ⊆ Sym(ps−1 n). Lemma 4.1. The map Φ : Sym(n) → Sym(ps−1 n) is a group monomorphism. Proof. We need to check that Φ(στ ) = Φ(σ)Φ(τ ) for all τ, σ ∈ Sym(n). Let i be a coordinate position in {1, . . . , ps−1 n} and χ(i) = ps−1 −(i mod ps−1 ). Note that χ(ps−1 a− χ(i)) = χ(i) for any integer a. Then     i + χ(i) (Φ(σ)Φ(τ ))(i) = Φ(σ) ps−1 τ − χ(i) ps−1 14 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 ⎛   ⎞ i+χ(i) ps−1 τ ps−1 − χ(i) + χ(i) = ps−1 σ ⎝ ⎠ − χ(i) ps−1   i + χ(i) = ps−1 στ − χ(i) ps−1 = Φ(στ )(i). Finally, it is easy to check that Φ is injective.  By Theorem 3.1, we identify PAut(Ht1 ,...,ts ) with the group π(L). Recall that we can label the ith coordinate position of Ht1 ,...,ts with the ith column wi of the generator matrix G t1 ,...,ts constructed via (3), i ∈ {1, . . . , n}. Any matrix M ∈ PAut(Ht1 ,...,ts ) sends columns of G t1 ,...,ts to other columns of G t1 ,...,ts . Therefore, M can be seen as a permutation of coordinate positions τ ∈ Sym(n), such that for all i ∈ {1, . . . , n} τ (i) = j ⇐⇒ wi M = wj , j ∈ {1, . . . , n}. (9) For any M ∈ PAut(Ht1 ,...,ts ), we define Φ(M) = Φ(τ ) ∈ Sym(ps−1 n) and, for any P ⊆ PAut(Ht1 ,...,ts ), we consider Φ(P) = {Φ(M) : M ∈ P} ⊆ Sym(ps−1 n). Proposition 4.2. Let Ht1 ,...,ts be the Zps -additive GH code of type (n; t1 , . . . , ts ) and let P ⊆ PAut(Ht1 ,...,ts ). Then, Φ(P) is an r-PD-set for H t1 ,...,ts with information set Φ(It1 ,...,ts ) if and only if for each r-set E of column vectors of G t1 ,...,ts there is M ∈ P such that {gM : g ∈ E} ∩ It1 ,...,ts = ∅. Proof. If Φ(P) is an r-PD-set with respect to the information set Φ(It1 ,...,ts ), then, for each r-set E ⊆ {1, . . . , ps−1 n}, there is τ ∈ P ⊆ Sym(n) such that Φ(τ )(E) ∩ Φ(It1 ,...,ts ) = ∅. For every r-set E ⊆ {1, . . . , n}, let Eo = {ps−1 (i − 1) + 1 : i ∈ E}. We know that there is τ ∈ P such that Φ(τ )(Eo ) ∩ Φ(It1 ,...,ts ) = ∅. By the definition of Φ, we also have that τ (E) ∩ It1 ,...,ts = ∅, which is equivalent to the statement. Conversely, we assume that for each r-set E ⊆ {1, . . . , n}, there is τ ∈ P ⊆ Sym(n) such that τ (E) ∩ It1 ,...,ts = ∅. For every r-set E ⊆ {1, . . . , ps−1 n}, let Eo ⊆ {1, . . . , n} be an r-set such that {i : ∃k ∈ {1, . . . , ps−1 } s.t. ϕk (i) ∈ E} ⊆ Eo , where ϕk (i) = ps−1 (i − 1) + k. Since there is τ ∈ P such that τ (Eo ) ∩ It1 ,...,ts = ∅, we have that Φ(τ )(E) ∩ Φ(It1 ,...,ts ) = ∅.  A slight modification of the proof of Proposition 4.2 leads to a more general result that holds for any Zps -additive code, not only for the family of Zps -additive GH codes. Proposition 4.3. Let C be a Zps -additive code, let I be an additive information set for C, and let S ⊆ PAut(C). Then, S satisfies that for each r-set E ⊆ {1, . . . , n} there is τ ∈ S such that τ (E) ∩ I = ∅ if and only if Φ(S) ⊆ PAut(C) satisfies that for each r-set E ⊆ {1, . . . , ps−1 n} there is σ ∈ Φ(S) such that σ(E) ∩ Φ(I) = ∅. A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 15 Definition 4.1. Let M ∈ PAut(Ht1 ,...,ts ) and let mi be the ith row of M, i ∈ {1, . . . , t1 + · · · + ts }. We define M∗ over Zps as the matrix where the first row is m1 and the ith row is m1 + mi for i ∈ {2, . . . , t1 }, m1 + pmi for i ∈ {t1 + 1, . . . , t1 + t2 }, m1 + p2 mi for i ∈ {t1 + t2 + 1, . . . , t1 + t2 + t3 } and so on until m1 + ps−1 mi for i ∈ {t1 + · · · + ts−1 + 1, . . . , t1 + · · · + ts }. Theorem 4.1. Let Ht1 ,...,ts be the Zps -additive GH code of type (n; t1 , . . . , ts ). Let Pr = {Mi : 0 ≤ i ≤ r} be a set of r + 1 matrices in PAut(Ht1 ,...,ts ). Then, Φ(Pr ) is an r-PD-set of size r + 1 for H t1 ,...,ts with information set Φ(It1 ,...,ts ) if and only if no two matrices (M−1 ∗ −1 ∗ i ) and (Mj ) have a row in common, for i, j ∈ {0, . . . , r} and i = j. Proof. The result can be proved using Proposition 4.2 and is a generalization of a similar result given in [2] for Z4 -linear Hadamard codes. However, we include the detailed proof for the convenience of the reader. Suppose that the set Pr = {Mi : 0 ≤ i ≤ r} satisfies that no two matrices (M−1 i ) ∗ −1 ∗ and (Mj ) , for i, j ∈ {0, . . . , r} and i = j, have a row in common. Assume that Φ(Pr ) is not an r-PD-set for H t1 ,...,ts with information set Φ(It1 ,...,ts ). By Proposition 4.2, it  follows that there exists an r-set E ⊆ {1} × Ztp1s−1 × (pZps ) 2 × · · · × ps−1 Zps , that t ts is, a set of r different column vectors of the generator matrix G t1 ,...,ts , such that for each i ∈ {0, . . . , r}, there is a gi ∈ E so that gi Mi ∈ It1 ,...,ts . Note that there are r + 1 values for i, but only r elements in E. Therefore, gMi ∈ It1 ,...,ts and gMj ∈ It1 ,...,ts for some g ∈ E and i = j. Suppose gMi = wh and gMj = wt , for wh , wt ∈ It1 ,...,ts . Then, g = wh M−1 i = wt M−1 j . Taking into account the form of the vectors in the information set It1 ,...,ts , by multiplying for such inverse matrices M−1 i and M−1 j , we obtain the first row or a certain addition between the first row and another row of each matrix. Thus, we obtain that (M−1 ∗ −1 ∗ i ) and (Mj ) have a row in common, contradicting our assumption. Let Pk ⊆ Pr of size k + 1. If this set satisfies the condition on the inverse matrices and we suppose that it is not a k-PD-set, we arrive to a contradiction in the same way as before. Conversely, suppose that Φ(Pr ) is an r-PD-set for H t1 ,...,ts with information set Φ(It1 ,...,ts ), but does not satisfy the condition on the inverse matrices. Thus, there are two matrices (M−1 ∗ −1 ∗ i ) and (Mj ) , with i, j ∈ {0, . . . , r}, such that they share a common row, say the hth row of (M−1 ∗ −1 ∗ i ) and the tth row of (Mi ) , with h, t ∈ {1, . . . , t1 + · · · + ts }. In other words, we can define g = eh (Mi ) = et (M−1 −1 ∗ ∗ j ) . Therefore, g = wh Mi −1 = −1 wt Mj , where wh , wt ∈ It1 ,...,ts . Finally, we obtain that gMi = wh and gMj = wt . Let L = { : 0 ≤  ≤ r,  = i, j}. For each  ∈ L, choose a row g of the matrix (M−1 ∗ −1 ∗ −1  ) . It is clear that g = eh (M ) = wh M , so g M = wh ∈ It1 ,...,ts . Finally, since some of the g may repeat, we obtain a set E = {g :  ∈ L} ∪ {g} of size at most r. Nevertheless, no matrix in Pr will map every member of E out of the additive information set It1 ,...,ts , which contradicts our assumption by Proposition 4.2.  16 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 Corollary 4.1. Let Pr be a set of r + 1 matrices in PAut(Ht1 ,...,ts ). If Φ(Pr ) is an r-PD- set of size r + 1 for H t1 ,...,ts , then any ordering of elements in Φ(Pr ) provides nested k-PD-sets for k ∈ {1, . . . , r}. Corollary 4.2. Let Pr be a set of r+1 matrices in PAut(Ht1 ,...,ts ). If Φ(Pr ) is an r-PD-set of size r + 1 for H t1 ,...,ts , then r ≤ fpt1 ,...,ts , where   pst1 +(s−1)t2 +···+ts −s − t1 − t2 − · · · − ts fpt1 ,...,ts = . t1 + t2 + · · · + ts Proof. Following the condition on sets of matrices to be r-PD-sets of size r + 1, given by Theorem 4.1, we have to obtain certain r +1 matrices with no rows in common. Since the rows of length t1 + · · · + ts must have 1 in the first coordinate, elements from Zps in the coordinates from 2 to t1 , and elements from pi Zps in the coordinates from t1 +· · ·+ti +1 to t1 + · · · + ti+1 , for i ∈ {1, . . . , s − 1}, the number of possible rows is ps(t1 −1)+(s−1)t2 +···+ts . Thus, taking this fact into account and counting the number of rows of each one of these r + 1 matrices, we have that (r + 1)(t1 + t2 + · · · + ts ) ≤ ps(t1 −1)+(s−1)t2 +···+ts , and the result follows.  5. Explicit construction of r-PD-sets of size r + 1 In this section, by using Theorem 4.1, we create r-PD-sets of size r + 1 for different infinite families of Zps -linear GH codes. First, we give an explicit construction for the Zps -linear GH codes H t1 ,0,...,0 , with t1 ≥ 2 and r ≤ fpt1 ,0,...,0 . Then, using a similar idea, we give an explicit construction for the Zps -linear GH codes Hi = H 1,t2 ,...,ts , where i ∈ {2, . . . , s}, tj = 0 for all j = i, ti ≥ 1, and r ≤ fp1,t2 ,...,ts . The main idea behind these constructions is to use a certain ordered set of vectors as rows of a set of matrices {N0∗ , . . . , Nr∗ }, such that {N0−1 , . . . , Nr−1 } is an r-PD-set. This method is a generalization of the one used in [2] for Z4 -linear Hadamard codes, which, in turn, was based on a similar idea for simplex codes given in [18]. Let R = GR(ps(t1 −1) ) be the Galois extension of dimension t1 − 1 over Zps , which is isomorphic to any ring Zps [x]/(h(x)), where h(x) is a monic basic irreducible polynomial over Zps of degree t1 − 1. A monic basic polynomial h(x) over Zps is called irreducible if h̄(x) is an irreducible polynomial over Zp , where h̄(x) is the polynomial obtained by taking the coefficients of h(x) modulo p. Moreover, if h̄(x) is primitive, then h(x) is said to be a monic basic primitive polynomial over Zps . If f (x) is an irreducible polynomial dividing xn − 1 in Zp [x], then there is a unique polynomial h(x) over Zps [x] that satisfies h̄(x) = f (x) and that divides xn − 1 in Zps [x], which is called the Hensel lift of f (x) to Zps . Moreover, if a polynomial of degree m is the Hensel lift of a monic primitive polynomial over Zp , then it always has a root of order pm − 1 [34]. Let h(x) be such a polynomial, with m = t1 − 1. Let α ∈ R be a root of h(x) of order  = pt1 −1 − 1. Then, the set T = {0, 1, α, α2 , . . . , α−1 } is called the Teichmüller set. A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 17 We can always represent an element y ∈ R in the following form: y = a1 + pa2 + p2 a3 + · · · + ps−1 as , where ai ∈ T , for i ∈ {1, . . . , s}, which is called the p-adic representation of y. Consider T as an ordered set. Then, we consider the following ordering of the elements of R = {y1 , . . . , yps(t1 −1) }: a1 + pa2 + · · · + ps−1 as < b1 + pb2 + · · · + ps−1 bs if aj < bj for the last j where aj and bj differ. We can also represent an element y ∈ R as a linear combination of some powers of α: y = b0 + b1 α + b2 α2 + · · · + bt1 −2 αt1 −2 , where bj ∈ Zps , for j ∈ {0, . . . , t1 − 2}. This is called the additive representation of y and it can be identified with the vector (b0 , b1 , . . . , bt1 −2 ) ∈ Ztp1s−1 . Using the ordering given by the p-adic representation, we construct the set {N0∗ , . . . , Nr∗ } of matrices of size t1 × t1 , where each one has the following form: ⎛ ⎞ 1 yt1 i+1 ⎜ ⎟ ⎜ 1 yt1 i+2 ⎟ ∗ ⎜ Ni = ⎜ . ⎟, .. ⎟ ⎝ .. . ⎠ 1 yt1 (i+1) with the elements yj , for j ∈ {1, . . . , ps(t1 −1) }, given as vectors of t1 − 1 components over Zps by using the corresponding additive representation. Note that no two matrices have a row in common, and there are |R|/t1  = fpt1 ,0,...,0 + 1 such matrices, where fpt1 ,0,...,0 = (pst1 − t1 )/t1  by Corollary 4.2, so r ≤ fpt1 ,0,...,0 . Let n∗i,j be the jth row of the matrix Ni∗ , for any i ∈ {0, . . . , r} and j ∈ {1, . . . , t1 }. In the context of the Zps -linear GH code H t1 ,0,...,0 , we define Ni as the matrix that has n∗i,1 as the first row and n∗i,j − n∗i,1 as the jth row, for j ∈ {2, . . . , t1 }. Note that this is consistent with Definition 4.1. Indeed, in the proof of Theorem 5.1, we see that N0 , . . . , Nr ⊆ PAut(Ht1 ,0,...,0 ). Lemma 5.1. Let K = Zp [x]/(f (x)), where f (x) ∈ Zp [x] is a primitive polynomial of degree m. Let α ∈ K be a root of f (x). Then, α − 1, α2 − 1, . . . , αm − 1 are linearly independent vectors over Zp . Proof. Let f (x) = xm +am−1 xm−1 +· · ·+a1 x +a0 , where aj ∈ Zp for all j ∈ {0, . . . , m − m−1 1}. Since α is a root of f (x), then αm − 1 = − j=0 aj αj − 1. Using the additive representation of the elements of K, we obtain the following vectors over Zm p : α −1 = i m−1 ei+1 − e1 for any i ∈ {1, . . . , m − 1}, and αm − 1 = − j=0 aj ej+1 − e1 . Consider the following m × m matrix over Zp by taking these vectors as rows: 18 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 −1 Idm−1 , −a0 − 1 −a where a = (a1 , . . . , am−1 ) ∈ Zm−1 . This matrix has the following determinant: m−1 p m−1 (−1)m ( j=0 aj + 1). Since f (x) is irreducible, then f (1) = 1 + j=0 aj = 0. Therefore, the determinant is non-zero and the vectors are linearly independent over Zp .  Theorem 5.1. Let Pr = {N0−1 , . . . , Nr−1 }. Then, Φ(Pr ) is an r-PD-set of size r + 1 for the Zps -linear GH code H t1 ,0,...,0 with information set Φ(It1 ,0,...,0 ), for all t1 ≥ 2 and 2 ≤ r ≤ fpt1 ,0,...,0 . Proof. By construction, the matrices N0∗ , . . . , Nr∗ do not share a row in common. Thus, if we prove that all matrices N0−1 , . . . , Nr−1 are in PAut(Ht1 ,0,...,0 ), then Φ(Pr ), where Pr = {N0−1 , . . . , Nr−1 }, would be an r-PD-set of size r + 1 for H t1 ,0,...,0 , by Theorem 4.1. Since PAut(Ht1 ,0,...,0 ) is a group, it is enough to prove that N0 , . . . , Nr are in this group. Note that these matrices are in the form ⎛ ⎞ 1 yt1 i+1 ⎜ ⎟ ⎜ 0 yt1 i+2 − yt1 i+1 ⎟ ⎜ Ni = ⎜ . ⎟, (10) .. ⎟ ⎝ .. . ⎠ 0 yt1 (i+1) − yt1 i+1 for any i ∈ {0, . . . , r}. As shown in (5), the elements in PAut(Ht1 ,0,...,0 ) have the form 1 a1 , (11) 0 A1,1 where a1 ∈ Ztp1s−1 and A1,1 ∈ GL(t1 − 1, Zps ). By using the additive representation, yy1 i+1 ∈ Ztp1s−1 , for any i ∈ {0, . . . , r}. Then, we need to prove that the vectors yt1 i+2 − yt1 i+1 , . . . , yt1 (i+1) − yt1 i+1 are linearly independent over Zps , for any i ∈ {0, . . . , r}. Taking into account that α = 1 and t1 ≤ pt1 −1 for t1 ≥ 2, the set of vectors {yt1 i+2 − yt1 i+1 , . . . , yt1 (i+1) − yt1 i+1 } is equal to one of the following three sets: L1 = {1, . . . , αt1 −2 }, L2 = {αk+1 − αk , . . . , αk+t1 −1 − αk }, L3 = {αk+1 − αk , . . . , α−1 − αk , −αk + pb, α − αk + pb, . . . , αk+t1 −2 − αk + pb}, for some k ∈ {0, . . . ,  − 1} and some b ∈ R. Clearly, L1 is a set of linearly independent vectors over Zps . t1 −1 For the second set L2 , suppose that i=1 λi (αk+i −αk ) = 0 for certain λi ∈ Zps , with  some of them being non-zero. Note that, since α is a unit in R, then i λi (αi − 1) = 0. Let m be the smallest integer in {0, . . . , s −1} for which there exists an i ∈ {1, . . . , t1 −1} A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 19 such that λi ∈ pm Zps and λi ∈ / pm+1 Zps . For example, if all λi ∈ pZps and there is a certain λi ∈ / p2 Z s , then m = 1. Therefore, we can define λi = pm λi for all i, and we   pi  obtain p m hence i λi (αi − 1) = ps−m λ for a certain λ ∈ R. Thus, by i λi (α − 1) = 0, taking modulo p, we obtain i λ̄i (ᾱi − 1) = 0 over Zp , with at least one λ̄i = 0. Clearly, ᾱ is a unit in Zp [x]/(h̄(x)). Therefore, by applying Lemma 5.1 on the vectors ᾱi − 1 for i ∈ {1, . . . , t1 − 1}, we obtain a contradiction. For the third set L3 , we follow a similar argument. Suppose that  −k−1 1 −2 t −λt1 −1 (α − pb) + k λi (α k+i −α )+ k λi (αk+i − αk + pb) = 0 i=1 i=−k for certain λi ∈ Zps , with some of them being non-zero. With the same definition −k−1 of m as in the previous case, we obtain −pm λt1 −1 (αk − pb) + pm i=1 λi (αk+i −  t1 −2 αk ) + pm i=−k λ (αk+i − αk + pb) = 0, where λi = pm λi . Thus, −λt1 −1 (αk − pb) + −k−1  k+i i k t1 −2  k+i λi (α − α ) + i=−k λi (α − αk + pb) = ps−m λ for some λ ∈ R. Taking i=1    modulo p, −λ̄t1 −1 ᾱ + i λ̄i (ᾱ k k+i − ᾱ ) = 0 over Zp , with at least one λ̄i = 0. Since k  ᾱ is a unit, we obtain −λ̄t1 −1 + i λ̄i (ᾱi − 1) = 0. We obtain a contradiction since the vectors −1, ᾱ − 1, . . . , ᾱt1 −2 − 1 are linearly independent over Zp .  Example 5.1. Let H3,0,0 be the Z27 -additive GH code of type (36 ; 3, 0, 0). Let R = GR(272 ) be the Galois ring over Z27 , isomorphic to Z27 [x]/(h(x)), where h(x) = x2 + 22x + 26. This polynomial can be obtained as the Hensel lift of f (x) = h̄(x) = x2 + x + 2 over Z3 . Note that h(x) is a monic basic primitive polynomial dividing x8 − 1 in Z27 [x]. Let α be a root of h(x) of order 8. Then, T = {0, 1, α, α2 , . . . , α7 } and we can order the elements of R as follows: R = {0 + 3 · 0 + 9 · 0, 1 + 3 · 0 + 9 · 0, α + 3 · 0 + 9 · 0, . . . , α7 + 3 · 0 + 9 · 0, 0 + 3 · 1 + 9 · 0, 1 + 3 · 1 + 9 · 0, α + 3 · 1 + 9 · 0, . . . , α7 + 3 · 1 + 9 · 0, ... 0 + 3 · α7 + 9 · α7 , 1 + 3 · α7 + 9 · α7 , α + 3 · α7 + 9 · α7 , . . . , α7 + 3 · α7 + 9 · α7 } = {0, 1, α, . . . , 22 + α, 3, 4, 3 + α, . . . , 25 + α, ... 21 + 12α, 22 + 12α, 21 + 13α, . . . , 16 + 13α}. By Theorem 5.1, we can find r-PD-sets of size r + 1 for all 2 ≤ r ≤ f33,0,0 = 242, by using the elements of R. Indeed, we can construct up to 243 matrices taking all the elements of R in groups of 3 in order to reach the upper bound. Here, we just show a smaller example by constructing an 11-PD-set formed by 12 matrices. 20 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 Consider the following 12 matrices, constructed by dividing the first 36 ordered ele- ments of R in groups of 3: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 1 1 5 1 0 26 ∗ ⎜ ⎟ ∗ ⎜ ⎟ ∗ ⎜ ⎟ N0 = ⎝ 1 1 0 ⎠ , N1 = ⎝ 1 5 26 ⎠ , N2 = ⎝ 1 26 22 ⎠ , 1 0 1 1 26 0 1 22 1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 3 0 1 4 5 1 3 26 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ N3∗ = ⎝ 1 4 0 ⎠ , N4∗ = ⎝ 1 8 26 ⎠ , N5∗ = ⎝ 1 2 22 ⎠ , 1 3 1 1 2 0 1 25 1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 3 1 1 8 1 0 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ N6∗ = ⎝ 1 1 3 ⎠ , N7∗ = ⎝ 1 5 2⎠, N8∗ = ⎝ 1 26 25 ⎠ , 1 0 4 1 26 3 1 22 4 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 3 15 1 4 20 1 3 14 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ N9∗ = ⎝ 1 4 15 ⎠ , N10 ∗ = ⎝1 8 14 ⎠ , ∗ N11 = ⎝1 2 10 ⎠ . 1 3 16 1 2 15 1 25 16 Note that there are no repeated rows in the whole set of matrices. Let P11 = {N0−1 , . . . , N11 −1 }, where ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 1 1 16 1 1 16 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ N0−1 = ⎝ 0 1 0 ⎠ , N1−1 = ⎝ 0 1 15 ⎠ , N2−1 = ⎝ 0 22 17 ⎠, 0 0 1 0 5 10 0 1 16 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 24 0 1 25 25 1 16 19 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ N3−1 = ⎝ 0 1 0 ⎠ , N4−1 = ⎝ 0 1 15 ⎠ , N5−1 = ⎝ 0 22 17 ⎠, 0 0 1 0 5 10 0 1 16 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 24 1 13 13 1 25 22 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ N6−1 = ⎝ 0 1 0 ⎠ , N7−1 = ⎝ 0 1 15 ⎠ , N8−1 = ⎝ 0 22 17 ⎠, 0 0 1 0 5 10 0 1 16 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 24 12 1 4 10 1 1 22 −1 ⎜ ⎟ −1 ⎜ ⎟ −1 ⎜ ⎟ N9 = ⎝ 0 1 0 ⎠ , N10 = ⎝ 0 1 15 ⎠ , N11 = ⎝ 0 22 17 ⎠. 0 0 1 0 5 10 0 1 16 The matrices of P11 can also be represented as permutations of coordinate positions as shown in (9). Let τi ∈ Sym(729) be the one corresponding to Ni−1 , i ∈ {0, . . . , 11}. Recall that Φ(Ni−1 ) = Φ(τi ) as defined in (8). Then, by Theorem 5.1, Φ(P11 ) = {Φ(Ni−1 ) : i ∈ {0, . . . , 11}} ⊆ Sym(6561) is an 11-PD-set of size 12 for the Z27 -linear GH code H 3,0,0 = Φ(H3,0,0 ) with information set Φ(I3,0,0 ) = {1, 2, 4, 10, 11, 13, 244, 245, 247}, given in Example 4.2. A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 21 Remark 5.1. By Corollary 4.2, fpt1 ,0,...,0 is the maximum number of errors that can be corrected using r-PD-sets of size r + 1 of the form Φ(Pr ), where Pr ⊆ PAut(Ht1 ,0,...,0 ). However, higher values of r could be achieved by considering elements in PAut(H t1 ,0,...,0 ) that are not the Φ image of elements in PAut(Ht1 ,0,...,0 ). Following a similar reasoning to the one used in Theorem 5.1, we can also obtain r-PD-sets of size r + 1 for the Zps -linear GH codes Hi = H 1,t2 ,...,ts , i ∈ {2, . . . , s}, of type (n; 1, t2 , . . . , ts ), where tj = 0 for all j = i, ti ≥ 1, and r ≤ fp1,t2 ,...,ts . Let Ri = GR(p(s−i+1)ti ) be the Galois extension of dimension ti over Zps−i+1 , isomorphic to Zps−i+1 [x]/(h(x)), with h(x) being a monic basic primitive polynomial of degree ti dividing xp −1 − 1 in Zps−i+1 [x]. Let α ∈ Ri be a root of h(x) of order  = pti − 1 ti and T = {0, 1, α, α2 , . . . , α−1 }. The p-adic representation of an element y ∈ Ri is y = a1 + pa2 + p2 a3 + · · · + ps−i as−i+1 , where ak ∈ T for k ∈ {1, . . . , s − i + 1}. Using this representation, we define the ordered set {y1 , . . . , yp(s−i+1)ti } with all the elements in Ri . Consider the set of matrices {M∗0 , . . . , M∗r } of size (ti + 1) × (ti + 1) over Zps , where ⎛ ⎞ 1 pi−1 y(ti +1)j+1 ⎜ ⎟ ⎜ 1 pi−1 y(ti +1)j+2 ⎟ ∗ ⎜ Mj = ⎜ . ⎟, .. ⎟ ⎝ .. . ⎠ 1 pi−1 y(ti +1)(j+1) for j ∈ {0, . . . , r}. Note that p(s−i+1)ti − 1 − ti r ≤ fp1,t2 ,...,ts =   1 + ti by Corollary 4.2. The elements y ∈ Ri are given as vectors over Zps−i+1 by using the additive representation. Then, we consider the inclusion of vectors y over Zps−i+1 to vectors over Zps as pi−1 y. Let (1, pi−1 m∗j,k ) be the kth row of the matrix M∗j , for any j ∈ {0, . . . , r}. In the context of the Zps -linear GH code Hi , we define Mj as the matrix that has (1, pi−1 m∗j,1 ) as the first row and (0, m∗j,k − m∗j,1 ) as the kth row, for k ∈ {2, . . . , ti + 1}. Note that this is consistent with Definition 4.1. Indeed, in the proof of Corollary 5.1, we see that M0 , . . . , Mr ⊆ PAut(Hi ), where Hi is the Zps -additive code such that Φ(Hi ) = Hi . Corollary 5.1. Let Pr = {M−1 −1 0 , . . . , Mr }. Then, Φ(Pr ) is an r-PD-set of size r + 1 for the Zps -linear GH code Hi = H 1,t2 ,...,ts , where ti ≥ 1, i ∈ {2, . . . , s}, and tj = 0 for all j ∈ {2, . . . , s} such that j = i, with information set Φ(I1,t2 ,...,ts ), for all 2 ≤ r ≤ fp1,t2 ,...,ts . Proof. Since Zps−i+1 is isomorphic to pi−1 Zps , the matrices M∗0 , . . . , M∗r do not share a row in common. Then, the matrix Mj is in the form 22 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 ⎛ ⎞ 1 pi−1 y(ti +1)j+1 ⎜ ⎟ ⎜ 0 y(ti +1)j+2 − y(ti +1)j+1 ⎟ ⎜ Mj = ⎜ . ⎟, (12) .. ⎟ ⎝ .. . ⎠ 0 y(ti +1)(j+1) − y(ti +1)j+1 for any j ∈ {0, . . . , r}. As shown in (5), the elements in PAut(H1,0,...,0,ti ,0,...,0 ) have the form 1 pi−1 ai , (13) 0 ζs−i+1 (Ai,i ) where ai ∈ Ztpis and Ai,i ∈ GL(ti , Zps ). Note that ζs−i+1 (Ai,i ) can be seen as an element of GL(ti , Zps−i+1 ) and, in fact, it can be any element in GL(ti , Zps−i+1 ). Following the same argument as in the proof of Theorem 5.1, over the Galois ring Ri = GR(p(s−i+1)ti ) instead of GR(ps(t1 −1) ), it can be proven that the vectors y(ti +1)j+2 − y(ti +1)j+1 , . . . , y(ti +1)(j+1) − y(ti +1)j+1 are linearly independent over Zps−i+1 .  Example 5.2. Let H1,3,0 be the Z8 -additive Hadamard code of type (26 ; 1, 3, 0). Let R2 = GR(43 ) be the Galois ring over Z4 , isomorphic to Z4 [x]/(h(x)), where h(x) = x3 + 2x2 + x + 3. This polynomial can be obtained as the Hensel lift of f (x) = h̄(x) = x3 + x + 1 over Z2 . Note that h(x) is a monic basic primitive polynomial dividing x7 − 1 in Z4 [x]. Let α be a root of h(x) of order 7. Then, T = {0, 1, α, α2 , . . . , α6 } and we can order the 64 elements of R2 as follows: R2 = {0 + 2 · 0, 1 + 2 · 0, α + 2 · 0, . . . , α6 + 2 · 0, 0 + 2 · 1, 1 + 2 · 1, α + 2 · 1, . . . , α6 + 2 · 1, ..., 0 + 2 · α 6 , 1 + 2 · α 6 , α + 2 · α 6 , . . . , α6 + 2 · α 6 } = {0, 1, α, . . . , 1 + 2α + α2 , 2, 3, 2 + α, . . . , 3 + 2α + α2 , ... 2 + 2α2 , 3 + 2α2 , 2 + α + 2α2 , . . . , 3 + 2α + 3α2 }. By Corollary 5.1, we can find r-PD-sets of size r + 1 for all 2 ≤ r ≤ f21,3,0 = 15, by using the elements of R2 . Indeed, we can construct up to 16 matrices taking all the elements of R2 , multiplied by 2, in groups of 4 in order to reach the upper bound. Here, we just show a smaller example by constructing an 8-PD-set formed by 9 matrices. Consider the following 9 matrices over Z8 , constructed by taking the first 36 ordered elements of R2 in groups of 4 and multiplying them by 2 as elements of Z38 : A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 23 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 0 1 2 6 4 1 4 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 2 0 0 ⎟ ⎜ 1 4 6 6 ⎟ ⎜ 1 6 0 0 ⎟ M∗0 = ⎜ ⎟ , M∗1 = ⎜ ⎟ , M∗2 = ⎜ ⎟, ⎝ 1 0 2 0 ⎠ ⎝ 1 6 6 2 ⎠ ⎝ 1 4 2 0 ⎠ ⎛ 1 0 0 2 ⎞ ⎛ 1 2 4 2 ⎞ ⎛ 1 4 0 2 ⎞ 1 6 6 4 1 0 4 0 1 2 2 4 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 0 6 6 ⎟ ⎜ 1 2 4 0 ⎟ ⎜ 1 4 2 6 ⎟ M∗3 = ⎜ ⎟ , M∗4 = ⎜ ⎟ , M∗5 = ⎜ ⎟, ⎝ 1 2 6 2 ⎠ ⎝ 1 0 6 0 ⎠ ⎝ 1 6 2 2 ⎠ ⎛ 1 6 4 2 ⎞ ⎛ 1 0 4 2 ⎞ ⎛ 1 2 0 2 ⎞ 1 0 0 4 1 2 6 0 1 4 4 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 2 0 4 ⎟ ⎜ 1 4 6 2 ⎟ ⎜ 1 6 4 0 ⎟ M∗6 = ⎜ ⎟ , M∗7 = ⎜ ⎟ , M∗8 = ⎜ ⎟. ⎝ 1 0 2 4 ⎠ ⎝ 1 6 6 6 ⎠ ⎝ 1 4 6 0 ⎠ 1 0 0 6 1 2 4 6 1 4 4 2 Note that there are no repeated rows in the whole set of matrices. Let P8 = {M−1 −1 0 , . . . , M8 }, where ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 0 1 6 4 6 1 4 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 3 3 0 ⎟ ⎜ 0 1 0 0 ⎟ M−1 = ⎜ ⎟ , M−1 = ⎜ ⎟ , M−1 = ⎜ ⎟, 0 ⎝ 0 0 1 0 ⎠ 1 ⎝ 0 2 3 3 ⎠ 2 ⎝ 0 0 1 0 ⎠ ⎛ 0 0 0 1 ⎞ ⎛ 0 2 1 0 ⎞ ⎛ 0 0 0 1 ⎞ 1 2 0 6 1 0 4 0 1 6 0 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 3 3 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 3 3 0 ⎟ M−1 3 =⎜ ⎟ , M−1 4 =⎜ ⎟ , M−1 5 =⎜ ⎟, ⎝ 0 2 3 3 ⎠ ⎝ 0 0 1 0 ⎠ ⎝ 0 2 3 3 ⎠ ⎛ 0 2 1 0 ⎞ ⎛ 0 0 0 1 ⎞ ⎛ 0 2 1 0 ⎞ 1 0 0 4 1 6 0 6 1 4 4 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 1 0 0 ⎟ ⎜ 0 3 3 0 ⎟ ⎜ 0 1 0 0 ⎟ M−1 = ⎜ ⎟ , M−1 = ⎜ ⎟ , M−1 = ⎜ ⎟. 6 ⎝ 0 0 1 0 ⎠ 7 ⎝ 0 2 3 3 ⎠ 8 ⎝ 0 0 1 0 ⎠ 0 0 0 1 0 2 1 0 0 0 0 1 The matrices of P8 can also be represented as permutations of coordinate positions as shown in (9). Let τi ∈ Sym(64) be the one corresponding to M−1 i , i ∈ {0, . . . , 8}. For −1 example, matrix M1 is equivalent to the permutation τ1 = (1, 60, 19, 56, 37, 46)(2, 55, 42, 23, 34, 63)(3, 50, 49, 54, 47, 16) (4, 61, 12, 21, 44, 29)(5, 38, 41, 28, 27, 32)(6, 33, 52, 59, 24, 45) (7, 48, 11, 26, 17, 62)(8, 43, 18, 57, 30, 15)(9, 20, 51, 64, 13, 14) (10, 31)(22, 39, 40, 35, 58, 25)(36, 53). Recall that Φ(M−1 j ) = Φ(τj ) as defined in (8). Then, by Corollary 5.1, Φ(P8 ) = {Φ(M−1 j ) : j ∈ {0, . . . , 8}} ⊆ Sym(256) is an 8-PD-set of size 9 for the Z8 -linear Hadamard code H 1,3,0 = Φ(H1,3,0 ) with information set Φ(I1,3,0 ) = {1, 2, 3, 5, 7, 17, 19, 65, 67}. 24 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 6. Recursive constructions of r-PD-sets In this section, given an r-PD-set of size  for a Zps -linear GH code H t1 ,...,ts , we show that one can easily obtain an r-PD-set of size  for the Zps -linear GH code H t1 +i1 ,...,ts +is , for all i1 , . . . , is ≥ 0. In particular, this is useful to obtain r-PD-sets for any code H t1 ,...,ts , including those of type different to (n; t1 , 0, . . . , 0) and (n; 1, 0, . . . , 0, ti , 0, . . . , 0), i ∈ {2, . . . , s}, which have been already considered in Section 5. We present two different constructions that produce a similar result. One uses the matrix representation of the elements in PAut(Ht1 ,...,ts ) and the other one uses the permutation representation. These constructions are a generalization of the ones given in [2] for Z4 -linear Hadamard codes. 6.1. Matrix representation In this first construction, we consider the elements of PAut(Ht1 ,...,ts ) as matrices in the subgroup π(L) of GL(t1 + · · · + ts , Zps ), described in Section 3. Consider a matrix M ∈ PAut(Ht1 ,...,ts ) in the form given in (5) and s positive integers κ1 , . . . , κs . Then, we define the matrix M(κ1 , . . . , κs ) as ⎛ ⎞ 1 a1 pa2 ··· ps−2 as−1 ps−1 as ⎜0 A1,1 pA1,2 ··· ps−2 A1,s−1 ps−1 A1,s ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 ζs−1 (A2,1 ) ζs−1 (A2,2 ) ··· s−3  s−2  ζs−1 (p A2,s−1 ) ζs−1 (p A2,s ) ⎟ ⎜. ⎟, ⎜. .. .. .. .. .. ⎟ ⎜. . . . . . ⎟ ⎜ ⎟ ⎝0 ζ2 (As−1,1 ) ζ2 (As−1,2 ) ··· ζ2 (As−1,s−1 ) ζ2 (pAs−1,s ) ⎠ 0 ζ1 (As,1 ) ζ1 (As,2 ) ··· ζ1 (As,s−1 ) ζ1 (As,s ) where a1 = (a1 , 0) ∈ Ztp1s−1+κ1 , aj = (aj , 0) ∈ Zpjs t +κj for j ∈ {2, . . . , s}, A1,1 = A1,1 0 Ai,i 0 ∈ GL(t1 − 1 + κ1 ), Ai,i = ∈ GL(ti + κi , Zps ) for 0 Idκ1 0 Idκi   Aj,i i ∈ {2, . . . , s}, and Ai,j = Ai,j 0 , Aj,i = are matrices over Zps for 0 i, j ∈ {1, . . . , s} with i < j, respectively. Note that M(κ1 , . . . , κs ) ∈ GL(t1 + · · · + ts + κ1 + · · · + κs , Zps ). Proposition 6.1. Let Pr = {M0 , . . . , Mr } ⊆ PAut(Ht1 ,...,ts ) such that Φ(Pr ) is an r-PD-set of size r + 1 for H t1 ,...,ts with information set Φ(It1 ,...,ts ). Then, Qr = {(M−1 i (κ1 , . . . , κs )) −1 : i ∈ {0, . . . , r}} ⊆ PAut(Ht1 +κ1 ,...,ts +κs ) and Φ(Qr ) is an r-PD- set of size r + 1 for H t1 +κ1 ,...,ts +κs with information set Φ(It1 +κ1 ,...,ts +κs ), for any κ1 , . . . , κs ≥ 0. Proof. Since M−1 i ∈ PAut(Ht1 ,...,ts ), M−1 i (κ1 , . . . , κs ) ∈ PAut(H t1 +κ1 ,...,ts +κs ), so Qr ⊆ PAut(H t1 +κ1 ,...,ts +κs ). Moreover, if Φ(Pr ) is an r-PD-set for H t1 ,...,ts , by Theorem 4.1, A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 25 the matrices (M−1 ∗ i ) for i ∈ {0, . . . , r} share no row in common. Clearly, the extended −1 matrices (Mi (κ1 , . . . , κs ))∗ do not share any row either.  6.2. Permutation representation In the second construction, the elements of PAut(Ht1 ,...,ts ) are considered as permu- tations in Sym(n), where n = ps(t1 −1)+(s−1)t2 +···+ts . Let σ ∈ Sym(n) and let q be a positive integer, then we define qσ ∈ Sym(qn) as the permutation that acts as σ in each of the following sets of coordinate positions: {1, . . . , n}, {n + 1, . . . , 2n}, {2n + 1, . . . , 3n}, . . . , {(q − 1)n + 1, . . . , qn}. Proposition 6.2. Let S be an r-PD-set of size  for H t1 ,...,ts of length n with information set I. Then, pS = {pσ : σ ∈ S} is an r-PD-set of size  for H t1 ,...,ts−1 ,ts +1 , with respect to any information set I  = I ∪ {j + n} with j ∈ I. Proof. Let H = Ht1 ,...,ts , H = Φ(H), H(s) = Ht1 ,...,ts−1 ,ts +1 and H (s) = Φ(H(s) ). Using the recursive construction given in (3), we obtain H (s) = {Φ((h, h, h, . . . , h) + λ(0, ps−1 , 2ps−1 , . . . , (p − 1)ps−1 )) : h ∈ H, λ ∈ Zp } = {(Φ(h), Φ(h + λps−1 ), Φ(h + λ2ps−1 ), . . . , Φ(h + λ(p − 1)ps−1 )) : h ∈ H, λ ∈ Zp }, where 0 and ps−1 are the vectors with 0 and ps−1 in all components, respectively. We have that Φ(h + λμps−1 ) = Φ(h) + λμΦ(ps−1 ) = Φ(h) + λμ1 for any λ, μ ∈ Zp [5]. Therefore, H (s) = {(h , h + λ1, h + λ2, . . . , h + λ(p − 1)) : h ∈ H, λ ∈ Zp }. If σ ∈ PAut(H), then σ(x) = y ∈ H for any x ∈ H. Consider an element x = (x, x + λ1, . . . , x + λ(p − 1)) ∈ H (s) . Then, (pσ)(x, x + λ1, . . . , x + λ(p − 1)) = (σ(x), σ(x) + σ(λ1), . . . , σ(x) + σ(λ(p − 1))) = (y, y + λ1, . . . , y + λ(p − 1)) ∈ H (s) , which means that pσ ∈ PAut(H (s) ). Let I ⊆ {1, . . . , n} be an information set for H. Define I  = I ∪ {j + n}, for any j ∈ I. We have that x|I  = (x|I , xj + λ), for any x = (x, x + λ1, . . . , x + λ(p − 1)) ∈ H (s) , where x ∈ H and λ ∈ Zp . Since there are pst1 +(s−1)t2 +···+ts different possible values of x|I and p possible values of λ, we obtain pst1 +(s−1)t2 +···+2ts−1 +ts +1 different elements x|I  , which means that I  is an information set for H (s) . Consider an error vector e = (e1 , . . . , ep ) ∈ Zpn p of weight wtH (e) ≤ r, where e = k (e1 , . . . , en ) ∈ Zp for k ∈ {1, . . . , p}. In order for pS to be an r-PD-set for H k k n (s) with 26 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 respect to I  , there must be an element pσ ∈ pS such that wtH ((pσ)(e)|I  ) = 0. Note that pσ(e) = (σ(e1 ), . . . , σ(ep )). Consider the vector ê = (ê1 , . . . , ên ) ∈ Znp such that êi = 1 if e1i > 0 or e2i > 0, and êi = 0 otherwise, i ∈ {1, . . . , n}. Since wtH (ê) ≤ r, there exists σ ∈ S such that wtH (σ(ê)|I ) = 0. Therefore, wtH (σ(e1 )|I ) = 0 and wtH (σ(e2 )|I ) = 0, hence wtH ((pσ(e))|I∪{j+n:j∈I} ) = 0. Since I  ⊆ I ∪ {j + n : j ∈ I}, we obtain that pS is an r-PD-set for H (s) with information set I  .  Note that Proposition 6.2 uses directly permutations from PAut(H t1 ,...,ts ), without as- suming that they come from elements in PAut(Ht1 ,...,ts ). This means that one can also use r-PD-sets for an r that may exceed the upper bound given by Corollary 4.2. This is not the case in the following proposition, since if S is an r-PD-set for H t1 ,...,ts , then it is not al- ways true that ps−i+1 S is an r-PD-set for H (i) for i ∈ {1, . . . , s−1}, where H (i) = Φ(H(i) ) and H(i) = Ht1 ,...,ti−1 ,ti +1,ti+1 ,...,ts . This is because if σ ∈ PAut(H t1 ,...,ts ), it is generally not true that ps−i+1 σ ∈ PAut(H (i) ). Instead, we have to assume that the r-PD-sets come from sets in PAut(Ht1 ,...,ts ) and extend each permutation σ ∈ PAut(Ht1 ,...,ts ) as ps−i+1 σ ∈ PAut(H(i) ) before applying the map Φ to obtain permutations in PAut(H (i) ). Proposition 6.3. Let S ⊆ PAut(Ht1 ,...,ts ) such that Φ(S) is an r-PD-set of size  for H t1 ,...,ts with information set I = Φ(I), where I is an additive information set for   Ht1 ,...,ts . Then, for any i ∈ {1, . . . , s}, Φ(ps−i+1 S) is an r-PD-set of size  for H t1 ,...,ts , with ti = ti + 1 and tj = tj for any j = i, with respect to any information set I  = Φ(I ∪ {j + n}) with j ∈ I, where n is the length of Ht1 ,...,ts . Proof. We follow a similar argument to the one given in Proposition 6.2, with the differ- ence that S is a subset of PAut(Ht1 ,...,ts ) and not of PAut(H t1 ,...,ts ). Let H = Ht1 ,...,ts ,   H = Φ(H), H(i) = Ht1 ,...,ts and H (i) = Φ(H(i) ). Taking into account that H(i) is con- structed using (3), H (i) = {(Φ(h), Φ(h + λpi−1 ), Φ(h + λ2pi−1 ), . . . , Φ(h + λ(ps−i+1 − 1)pi−1 )) : h ∈ H, λ ∈ Zps−i+1 }. If τ ∈ PAut(H), then (ps−i+1 τ )(h) = (σ(h), σ(h) + σ(λpi−1 ), . . . , σ(x) + σ(λ(ps−i+1 − 1)pi−1 )) = (σ(h), σ(h) + λpi−1 , . . . , σ(h) + λ(ps−i+1 − 1)pi−1 ) ∈ H(i) , for any h = (h, h +λpi−1 , . . . , h +λ(ps−i+1 −1)pi−1 ) ∈ H(i) , with h ∈ H and λ ∈ Zps−i+1 . Therefore, (ps−i+1 τ ) ∈ PAut(H(i) ) and Φ(ps−i+1 τ ) ∈ PAut(H (i) ). By Proposition 4.1, the set I ∪ {n + 1} is an additive information set for H(i) . In fact, in the proof we also show that any set I ∪ {x}, for x ∈ {n + 1, . . . , 2n} is also an information set. In particular I  = I ∪ {j + n}, for any j ∈ I, is an information set A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 27 for H(i) and I  = Φ(I  ) is an information set for H (i) . Note that I  has s − i + 1 more coordinates than I. s−i+1 Finally, consider an error vector e = (e1 , . . . , ep ) of weight wtH (e) ≤ r, where ek = (ek1 , . . . , ekn ) ∈ Znp for k ∈ {1, . . . , ps−i+1 }. Define the vector ê = (ê1 , . . . , ên ) ∈ Znp that satisfies êm = 1 if e1m = 0 or e2m = 0, and êm = 0 otherwise, m ∈ {1, . . . , n}. Since wtH (ê) ≤ r, there exists τ ∈ S such that wtH (Φ(τ )(ê)|I ) = 0. Thus, wtH (Φ(τ )(e1 )|I ) = s−i+1 0 and wtH (Φ(τ )(e2 )|I ) = 0. Note that Φ(ps−i+1 τ )(e) = (Φ(τ )(e1 ), . . . , Φ(τ )(ep )). Therefore, wtH (Φ(ps−i+1 τ )(e)|I∪{j+ps−1 n:j∈I} ) = wtH (Φ(τ )(e1 )|I ) + wtH (Φ(τ )(e2 )|I ) = 0. Since I  ⊆ I ∪ {j + ps−1 n : j ∈ I}, this implies that Φ(ps−i+1 S) is an r-PD-set for H (i) with information set I  .  Remark 6.1. By the definition of ps−i+1 τ and Φ, we have that Φ(ps−i+1 τ ) = ps−i+1 Φ(τ ), for any τ ∈ PAut(Ht1 ,...,ts ) and i ∈ {1, . . . , s}. By the proof of Proposition 6.3, ps−i+1 τ ∈ PAut(H(i) ), so ps−i+1 Φ(τ ) ∈ PAut(H (i) ), where H(i) , H (i) are defined as in this proof. Example 6.1. Consider the Z27 -linear GH code H 3,0,0 as in Example 5.1. We have that Φ(P11 ) is an 11-PD-set of size 12, where P11 can be identified by the set of permutations {id, τ1 , . . . , τ11 } ⊆ Sym(729). Then, by Proposition 6.2 or Proposition 6.3, we know that the following subset of Sym(19683): {3Φ(id), 3Φ(τ1 ), . . . , 3Φ(τ11 )} is an 11-PD-set for the Z27 -linear GH code H 3,0,1 , with information set Φ(I3,0,1 ) = Φ(I3,0,0 ) ∪ Φ(3) ({730}) = {1, 2, 4, 10, 11, 13, 244, 245, 247, 6562}. Similarly, by Proposi- tion 6.3 and Remark 6.1, we know that the following subset of Sym(59049): {9Φ(id), 9Φ(τ1 ), . . . , 9Φ(τ11 )} is an 11-PD-set for the Z27 -linear GH code H 3,1,0 , with information set Φ(I3,1,0 ) = Φ(I3,0,0 ) ∪ Φ(2) ({730}) = {1, 2, 4, 10, 11, 13, 244, 245, 247, 6562, 6565}. In general, we can construct r-PD-sets for H 3,0,1 and H 3,1,0 for any r ≤ f33,0,0 = 242. We could also use Proposition 6.3 in order to obtain an 11-PD-set for the Z27 -linear GH code H 4,0,0 , or in general an r-PD-set for any r ≤ f33,0,0 = 242. However, in this case, we can construct an r-PD-set directly, from the explicit construction presented in Section 5, for any r ≤ f34,0,0 = 4919. Corollary 6.1. Let S ⊆ PAut(Ht1 ,...,ts ) such that Φ(S) is an r-PD-set of size  for H t1 ,...,ts with information set I. Then, Φ(psi1 +(s−1)i2 +···+is S) is an r-PD-set of size  for H t1 +i1 ,t2 +i2 ,...,ts +is , with the information set obtained by applying recursively Propo- sition 4.1, for any i1 , i2 , . . . , is ≥ 0. 28 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 Corollary 6.2. If Pr = {N0−1 , . . . , Nr−1 }, as defined in Section 5 for the Zps -linear GH code H t1 ,0,...,0 , then Φ(p(s−1)t2 +···+ts Pr ) is an r-PD-set of size r + 1 for the Zps -linear GH code H t1 ,...,ts , for all t2 , . . . , ts ≥ 0, t1 ≥ 2 and 2 ≤ r ≤ fpt1 ,0,...,0 . Similarly, if Pr = {M−1 −1 0 , . . . , Mr }, as defined in Section 5 for the Zps -linear GH code H 1,0,...,0,ti ,0,...,0 . Then, Φ(p Pr ), where a = s(t1 − 1) + · · · + (s − i + 2)ti−1 + (s − i)ti+1 + · · · + ts , is an a r-PD-set of size r +1 for the Zps -linear GH code H t1 ,...,ts , for all t2 , . . . , ts ≥ 0, t1 , ti ≥ 1 and 2 ≤ r ≤ fp1,0,...,0,ti ,0,...,0 . Depending on the type of the Zps -linear GH code, the largest r allowed by Corol- lary 6.2 may be either fpt1 ,0,...,0 or one of fp1,0,...,0,ti ,0,...,0 , for i ∈ {2, . . . , s}. Let us define f˜pt1 ,...,ts = max{fpt1 ,0,...,0 , fp1,t2 ,0,...,0 , . . . , fp1,0,...,0,ts } ≤ fpt1 ,...,ts . If f˜pt1 ,...,ts = fpt1 ,0,...,0 , we achieve the largest r by using the explicit construction to obtain Pr for H t1 ,0...,0 and then extending the r-PD-set as Φ(p(s−1)t2 +···+ts Pr ). However, if f˜pt1 ,...,ts = fp1,0,...,0,ti ,0,...,0 , we achieve the largest r by using the ex- plicit construction to obtain Pr for H 1,0,...,0,ti ,0,...,0 and then extending the PD-set as Φ(ps(t1 −1)+···+(s−i+2)ti−1 +(s−i)ti+1 +···+ts Pr ). Example 6.2. Consider the Z8 -linear Hadamard codes H 3,3,7 , H 3,4,7 and H 3,3,8 . By Corollary 4.2, we have f23,0,0 = 20, f21,3,0 = 15, f21,0,7 = 15, f21,4,0 = 50, f21,0,8 = 27. Therefore, for example, • for H 3,3,7 , since f˜23,3,7 = max{f23,0,0 , f21,3,0 , f21,0,7 } = f23,0,0 = 20, it is better to start by using the explicit construction for H 3,0,0 . • for H 3,4,7 , since f˜23,4,7 = f21,4,0 = 50, it is better to start with the explicit construction for H 1,4,0 . • for H 3,3,8 , since f˜23,3,8 = f21,0,8 = 27, it is better to start with the explicit construction for H 1,0,8 . 7. Computational results The explicit constructions presented in Section 5 give r-PD-sets of size r + 1 with an r that reaches up to the upper bound given by Corollary 4.2. However, these constructions are only defined for some specific Zps -linear GH codes: H t1 ,0,...,0 and H 1,0,...,0,ti ,0,...,0 , with t1 ≥ 2 and ti ≥ 1, respectively. The recursive constructions presented in Section 6 allow to obtain r-PD-sets for all Zps -linear GH codes H t1 ,...,ts , but they may not achieve the upper bound. Indeed, for the codes where the explicit constructions can not be applied, r ≤ f˜pt1 ,...,ts < fpt1 ,...,ts , so other strategies are necessary in order to achieve a value of r closer to the theoretical upper bound fpt1 ,...,ts . In this section, we present some computational results, obtained by using the computer algebra system Magma [9]. These results show that we can increase the value of r for Zps - linear GH codes H t1 ,...,ts , by looking for r-PD-sets randomly. We follow a similar method A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 29 Table 1 Maximum value r for which r- PD-sets were found for some codes H t1 ,t2 , with p = 2, us- ing a non-deterministic method. Comparison with previous re- sults, rold , given in [2] and the upper bound f2t1 ,t2 . t1 t2 rold r f2t1 ,t2 3 0 4 4 4 1 6 7 7 2 10 11 11 3 16 18 20 4 26 31 35 5 42 50 63 4 0 15 15 15 1 23 23 24 2 36 38 41 3 56 62 72 4 91 103 127 5 150 172 226 5 0 50 50 50 1 72 76 84 2 116 124 145 3 187 199 255 4 312 321 454 5 518 551 818 as the one used in [2]. That is, we generate sets Pr = {M0 , . . . , Mr } of r + 1 random matrices in PAut(Ht1 ,...,ts ) such that all rows from the matrices of {M∗0 , . . . , M∗r } are different. The sets are constructed incrementally, starting from different initial matrices M0 until the target value of r is achieved. Initially, the target value of r is defined as the upper bound fpt1 ,...,ts . If the method has generated k < r matrices M∗0 , . . . , M∗k , and fails to generate M∗k+1 in a defined time constraint, then it starts again from another initial matrix M∗0 . If the target value r is not attained after a certain number of different initial matrices, then r is decreased by one and the process starts again. If, by decreasing r, it reaches the value of f˜pt1 ,...,ts , then the r-PD-set given by Corollary 6.2 is returned. Table 1 shows the maximum values of r obtained for Z4 -linear Hadamard codes H t1 ,t2 , with 3 ≤ t1 ≤ 5 and 0 ≤ t2 ≤ 5. They are compared with the values given in [2] and the upper bound f2t1 ,t2 . The results from [2] were obtained by using a method that is currently implemented in the Magma function PDSetHadamardCodeZ4(t1, t2: AlgMethod:= "Nondeterministic") included in the official distribution [9]. We have corrected an error found in the implementation of this function and made some im- provements, which has allow us to achieve larger values of r in this case. Then, we have generalized these functions to deal with Zps -linear GH codes. Table 2 shows the maxi- mum values of r obtained for Z8 -linear Hadamard codes H t1 ,t2 ,t3 , with t1 = 3, 0 ≤ t2 ≤ 2 and 0 ≤ t3 ≤ 3. The upper bounds f˜23,t2 ,t3 and f23,t2 ,t3 are also shown in order to see the improvement with respect to the recursive construction, which is bounded by f˜23,t2,t3 , and with respect to the theoretical maximum, given by f23,t2 ,t3 . 30 A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 Table 2 Maximum value r for which r-PD-sets were found for some codes H 3,t2 ,t3 , with p = 2, using a non-deterministic method. Comparison with the upper bound of the recursive constructions f˜23,t2 ,t3 and the upper bound f23,t2 ,t3 . t2 t3 f˜23,t2 ,t3 r f23,t2 ,t3 0 0 20 20 20 1 20 30 31 2 20 46 50 3 20 73 84 1 0 20 61 63 1 20 94 101 2 20 149 169 3 20 242 291 2 0 20 189 203 1 20 299 340 2 20 476 584 3 20 773 1023 The Magma function developed to construct r-PD-sets of size r + 1 for Zps -linear GH codes has been included in a new Magma package to deal with linear codes over Zps [17]. This package also allows the construction of Zps -linear GH codes, and includes functions related to generalized Gray maps, information sets, the process of encoding and decoding using permutation decoding, among others. This package generalizes some of the functions for codes over Z4 , which are already included in the standard Magma distribution [9]. It has been developed mainly by the authors of this paper and the collaboration of some undergraduate students. The first version of this new package and a manual describing all functions will be released this year, and it will be available in a GitHub repository and in the CCSG web site (http://ccsg.uab.cat) 8. Conclusions In this paper, we determine the permutation automorphism group of Zps -additive GH codes, PAut(Ht1 ,...,ts ), and give a representation of the elements as matrices of the general linear group over Zps of dimension t1 + · · · + ts . Then, explicit constructions of r-PD-sets of size r+1 for Zps -linear GH codes of types (n; t1 , 0, . . . , 0) and (n; 1, 0, . . . , 0, ti , 0, . . . , 0), with t1 ≥ 2 and ti ≥ 1, respectively, are given. For these cases, the value of r is upper- bounded by fpt1 ,0,...,0 or fp1,0,...,0,ti ,0,...,0 depending on the type. In general, for Zps -linear GH codes of any type (n; t1 , . . . , ts ), we also present some constructions of r-PD-sets of size r + 1, but only up to r ≤ f˜pt1 ,...,ts ≤ fpt1 ,...,ts . The computational results given in Section 7 confirm that r-PD-sets of size r + 1, with values of r closer to the theoretical upper bound fpt1 ,...,ts , may exist for Zps -linear GH codes of any type. Therefore, a natural further research on this topic would be to find explicit constructions for codes H t1 ,...,ts , with f˜pt1 ,...,ts ≤ r ≤ fpt1 ,...,ts . A. Torres-Martín, M. Villanueva / Finite Fields and Their Applications 93 (2024) 102316 31 Another direction which extends this line of research would be the generalization of these results to Zp Zp2 · · · Zps -linear GH codes, which are GH codes and can be obtained p ×Zp2 ×· · ·× α2 from the generalized Gray map image of subgroups over mixed alphabets Zα 1 αs Zps . In particular, Z2 Z4 -linear codes have been studied extensively, see for example [7,8], and the permutation decoding method given in [4] is also defined for these codes, since they are systematic. More generally, Zp Zp2 · · · Zps -linear codes have been studied for example in [1,31]. The results given in [32] can be extended to Zp Zp2 · · · Zps -linear codes, in order to obtain a systematic encoding for Zp Zp2 · · · Zps -linear codes, which allow us to use the permutation decoding method for these codes. This gives a motivation to construct r-PD-sets for Zp Zp2 · · · Zps -linear GH codes, which have been recently studied in [5,6] showing that they are not necessarily equivalent to the Zps -linear GH codes considered in this paper. For any Zps -additive GH code Ht1 ,...,ts , we have obtained PAut(Ht1 ,...,ts ), which is a subgroup of the monomial automorphism group, MAut(Ht1 ,...,ts ). We also have that Φ(PAut(Ht1 ,...,ts )) ⊆ PAut(H t1 ,...,ts ). The study of these groups, MAut(Ht1 ,...,ts ) and PAut(H t1 ,...,ts ), also remain as an open problem for p ≥ 3 or s ≥ 3. For Z4 -linear Hadamard codes, these groups are studied in [25]. 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