Partial permutation decoding for binary linear and Z4-linear Hadamard codes ∗ Roland D. Barrolleta† Mercè Villanueva† arXiv:1512.01839v2 [cs.IT] 30 Apr 2016 September 18, 2021 Abstract Permutation decoding is a technique which involves finding a subset S, called PD-set, of the permutation automorphism groupj of a codek C in m −m−1 -PDorder to assist in decoding. An explicit construction of 2 1+m j m k 2 −m−1 sets of minimum size + 1 for partial permutation decoding for 1+m binary linear Hadamard codes Hm of length 2m , for all m ≥ 4, is described. Moreover, a recursive construction to obtain s-PD-sets of size l for Hm+1 of length 2m+1 , from a given s-PD-set of the same size for Hm , is also established. These results are generalized to find s-PD-sets for (nonlinear) binary Hadamard codes of length 2m , called Z4 -linear Hadamard codes, which are obtained as the Gray map image of quaternary linear codes of length 2m−1 . Index terms— automorphism group, permutation decoding, PD-set, Hadamard code, Z4 -linear code 1 Introduction Denote by Z2 and Z4 the rings of integers modulo 2 and modulo 4, respectively. Let Zn2 denote the set of all binary vectors of length n and let Zn4 be the set of all n-tuples over the ring Z4 . The Hamming weight wt(v) of a vector v ∈ Zn2 is the number of nonzero coordinates in v. The Hamming distance d(u, v) between two vectors u, v ∈ Zn2 is the number of coordinates in which u and v differ, that is, d(u, v) = wt(u+v). Let ei be the binary vector or tuple over Z4 with a one in the ith coordinate and zeros elsewhere. Let 0, 1, 2 and 3 be the binary vectors or tuples over Z4 having 0, 1, 2 and 3, respectively, repeated in each coordinate. It will be clear by the context whether we refer to binary vectors or tuples over Z4 . Any nonempty subset C of Zn2 is a binary code and a subgroup of Zn2 is called a binary linear code. Equivalently, any nonempty subset C of Zn4 is a quaternary ∗ This work was partially supported by the Spanish MINECO under Grant TIN2013-40524P, and by the Catalan AGAUR under Grant 2014SGR-691. The material in this paper was presented in part at IX “Jornadas de Matemática Discreta y Algorı́tmica” in Tarragona, Spain, 2014 [1]. † Departament d’Enginyeria de la Informació i de les Comunicacions, Universitat Autònoma de Barcelona, e-mails: rolanddavid.barrolleta@uab.cat and merce.villanueva@uab.cat. 1 code and a subgroup of Zn4 is called a quaternary linear code. Quaternary codes can be seen as binary codes under the usual Gray map Φ : Zn4 → Z2n 2 defined as Φ((y1 , . . . , yn )) = (φ(y1 ), . . . , φ(yn )), where φ(0) = (0, 0), φ(1) = (0, 1), φ(2) = (1, 1), φ(3) = (1, 0), for all y = (y1 , . . . , yn ) ∈ Zn4 . If C is a quaternary linear code, the binary code C = Φ(C) is said to be a Z4 -linear code. Moreover, since C is a subgroup of Zn4 , it is isomorphic to an abelian group Zγ2 × Zδ4 and we say that C (or equivalently the corresponding Z4 -linear code C = Φ(C)) is of type 2γ 4δ [7]. Let C be a binary code of length n and size |C| = 2k . For a vector v ∈ Zn2 and a set I ⊆ {1, . . . , n}, we denote by vI the restriction of v to the coordinates in I and by CI the set {vI : v ∈ C}. A set I ⊆ {1, . . . , n} of k coordinate positions is an information set for C if |CI | = 2k . If such an I exists, C is said to be a systematic code. For each information set I of size k, the set {1, . . . , n}\I of the remaining n − k coordinate positions is a check set for C. Let Sym(n) be the symmetric group of permutations on the set {1, . . . , n} and let id ∈ Sym(n) be the identity permutation. The group operation in Sym(n) is the function composition σ1 σ2 , which maps any element x to σ1 (σ2 (x)), σ1 , σ2 ∈ Sym(n). A σ ∈ Sym(n) acts linearly on words of Zn2 or Zn4 by permuting their coordinates as follows: σ((v1 , . . . , vn )) = (vσ−1 (1) , . . . , vσ−1 (n) ). The permutation automorphism group of C or C = Φ(C), denoted by PAut(C) or PAut(C), respectively, is the group generated by all permutations that preserve the set of codewords. A binary Hadamard code of length n has 2n codewords and minimum distance n/2. It is well-known that there exists an unique binary linear Hadamard code Hm of length n = 2m , for any m ≥ 2. The quaternary linear codes such that, under the Gray map, give a binary Hadamard code are called quaternary linear Hadamard codes and the corresponding Z4 -linear codes are called Z4 -linear Hadamard codes. These codes have been studied and classified in [10, 13], and their permutation automorphism groups have been determined in [9, 14]. Permutation decoding is a technique, introduced in [11] by MacWilliams for linear codes, which involves finding a subset of the permutation automorphism group of a code in order to assist in decoding. A new permutation decoding method for Z4 -linear codes (not necessarily linear) was introduced in [2]. In general, the method works as follows. Given a systematic t-error-correcting code C with information set I, we denote by y = x + e the received vector, where x ∈ C and e is the error vector. Suppose that at most t errors occur, that is, wt(e) ≤ t. The permutation decoding consists on moving all errors in y out of I, by using an automorphism of C. This technique is strongly based on the existence of some special subsets of PAut(C), called PD-sets. Specifically, a subset S ⊆ PAut(C) is said to be an s-PD-set for the code C if every s-set of coordinate positions is moved out of I by at least one element of S, where 1 ≤ s ≤ t. When s = t, S is said to be a PD-set. In [4], it is shown how to find s-PD-sets of size s + 1 that satisfy the GordonSchönheim bound for partial permutation decoding for  the binary simplex code  m . In this paper, following of lenght 2m − 1 for all m ≥ 4 and 1 < s ≤ 2 −m−1 m the same technique, similar results for binary linear and Z4 -linear Hadamard
codes are established. In [15], 2-PD-sets of size 5 and 4-PD-sets of size m+1 2 + 2 are found for binary linear Hadamard codes Hm , for all m > 4. Small PDsets that satisfy the Gordon-Schönheim bound have also been found for binary 2 Golay codes [5, 16] and for the binary simplex code S4 [8]. This work is organized as follows. In Section 2, we prove that the GordonSchönheim bound can be adapted to systematic codes, not necessarily linear. Furthermore, we apply this bound on the minimum size of s-PD-sets to binary linear and Z4 -linear Hadamard codes, which are systematic but nonlinear in general, and we prove that their minimum size is s + 1. In Section 3, we regard the permutation automorphism group PAut(Hm ) as a certain subgroup of the general linear group GL(m + 1, 2) and we provide a criterion on subsets of matrices of such subgroup to be an s-PD-set of size s + 1 for Hm . In Section 4, we define recursive constructions to obtain s-PD-sets of size l for Hm+1 from a given s-PD-set of the same size for Hm , where l ≥ s + 1. Finally, in Sections 5 and 6, we establish equivalent results for (nonlinear) Z4 -linear Hadamard codes. 2 Minimum size of s-PD-sets for Hadamard codes There is a well-known bound on the minimum size of PD-sets for linear codes based on the length, dimension and minimum distance of such codes that can be adapted to systematic codes (not necessarily linear) easily. Proposition 1. Let C be a systematic t-error correcting code of length n, size |C| = 2k and minimum distance d. Let r = n − k be the redundancy of C. If S is a PD-set for C, then       n−t+1 n n−1 ... ... . (1) |S| ≥ r r−1 r−t+1 The above inequality (1) is often called the Gordon-Schönheim bound. The result given by Proposition 1 is quoted and proved for linear codes in [6]. We can follow the same proof, since the linearity of the code is only used to guarantee that the code is systematic. In [2], it is shown that Z4 -linear codes are systematic, and a systematic encoding is given for these codes. Therefore, the result can be applied to any Z4 -linear code, not necessarily linear. The Gordon-Schönheim bound can be adapted to s-PD-sets for all s up to the error correcting capability of the code. Note that the error-correcting capability of any binary linear or Z4 -linear Hadamard code of length n = 2m is  m−1 tm = ⌊(d − 1)/2⌋ = (2 − 1)/2 = 2m−2 − 1 [12]. Moreover, all these codes are systematic and have size 2n = 2m+1 . Therefore, the right side of the bound given by (1), for binary linear and Z4 -linear Hadamard codes of length 2m and for all 1 ≤ s ≤ tm , becomes     m   2m 2m − 1 2 −s+1 . . . ... . (2) gm (s) = m 2 − m − 1 2m − m − 2 2m − m − s We compute the minimum value of gm (s) in the following lemma. Lemma 2. Let m be an integer, m ≥ 4. For 1 ≤ s ≤ tm ,     m   2m 2m − 1 2 −s+1 gm (s) = m . . . . . . ≥ s + 1, 2 − m − 1 2m − m − 2 2m − m − s where tm = 2m−2 − 1 is the error-correcting capability of any binary linear and Z4 -linear Hadamard code of length 2m . 3 Proof. We need to prove that gm (s) ≥ s + 1. This fact is clear, since the central term  m  2 −s+1 =2 2m − m − s for all s ∈ {1, . . . , 2m−2 − 1}, and in each stage of the ceiling function working from inside, gm (s) increases its value by at least 1. The smaller the size of the PD-set is, the more efficient permutation decoding becomes. Because of this, we will focus on the case when we have that gm (s) = s + 1. For each binary linear and Z4 -linear Hadamard code of length 2m , m ≥ 4, we define the following integer: fm = max{s : 2 ≤ s, gm (s) = s + 1}, which represents the greater s in which we can find s-PD-sets of size s + 1. The following result characterize this parameter from the value of m. Note that for m = 3, since the error-correcting capability is t3 = 1, the permutation decoding becomes unnecessary and we do not take it into account in the results. k j m −m−1 . Lemma 3. Let m be an integer, m ≥ 4. Then, fm = 2 1+m Proof. The result is easy to prove by Lemma 2 and following a similar argument as the one in the proof of Lemma 2 in [4]. 3 Finding s-PD-sets of size s + 1 for binary linear Hadamard codes For any m ≥ 2, there is an unique binary linear Hadamard code Hm of length 2m [12]. A generator matrix Gm for Hm can be constructed as follows:   1 1 , (3) Gm = 0 G′ where G′ is any matrix having as column vectors the 2m − 1 nonzero vectors from Zm 2 , with the vectors ei , i ∈ {1, . . . , m}, in the first m positions. Note that G′ can be seen as a generator matrix of the binary simplex code of length 2m − 1. By construction, from (3), it is clear that Im = {1, . . . , m + 1} is an information set for Hm . Let wi be the ith column vector of Gm , i ∈ {1, . . . , 2m }. By labelling the coordinate positions with the columns of Gm , we can take as an information set Im for Hm the first m + 1 column vectors of Gm considered as row vectors, that is, Im = {w1 , . . . , wm+1 } = {e1 , e1 + e2 , . . . , e1 + em+1 }. Then, depending on the context, Im will be taken as a subset of {1, . . . , 2m } or as a subset of {1} × Zm 2 . Example 4. Let generator matrix  1  0  G4 =   0  0 0 H4 be the binary linear Hadamard code of length 16 with 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 0 1 1 4 1 1 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1    ,   (4) constructed as in (3). The set I4 = {1, 2, 3, 4, 5}, or equivalently the set of column vectors I4 = {w1 , w2 , w3 , w4 , w5 } = {e1 , e1 + e2 , e1 + e3 , e1 + e4 , e1 + e5 } of G4 , is an information set for H4 . It is known that the permutation automorphism group PAut(Hm ) of Hm is isomorphic to the general affine group AGL(m, 2) [12]. Let GL(m, 2) be the general linear group over Z2 . Recall that AGL(m, 2) consists of all mappings m m α : Zm 2 → Z2 of the form α(x) = Ax + b for x ∈ Z2 , where A ∈ GL(m, 2) and m b ∈ Z2 , together with the function composition as the group operation. The monomorphism ϕ: AGL(m, 2) −→ (b, A) 7−→ GL(m + 1,2)  1 b 0 A defines an isomorphism between AGL(m, 2) and the subgroup of GL(m + 1, 2) consisting of all nonsingular matrices whose first column is e1 . Therefore, from now on, we also regard PAut(Hm ) as this subgroup. Note that any matrix M ∈ PAut(Hm ) can be seen as a permutation of coordinate positions, that is, as an element of Sym(2m ). By multiplying each column vector wi of Gm by M , we obtain another column vector wj = wi M , which means that the ith coordinate position moves to the jth coordinate position, i, j ∈ {1, . . . , 2m }. Let M ∈ PAut(Hm ) and let mi be the ith row of M , i ∈ {1, . . . , m + 1}. We define M ∗ as the matrix where the first row is m1 and the ith row is m1 + mi , i ∈ {2, . . . , m+1}. An s-PD-set of size s+1 for Hm meets the Gordon-Schönheim bound if 2 ≤ s ≤ fm . The following theorem provides us a condition on sets of matrices of PAut(Hm ) in order to be s-PD-sets of size s + 1 for Hm . Theorem 5. Let Hm be the binary linear Hadamard code of length 2m , with m ≥ 4. Let Ps = {Mi : 0 ≤ i ≤ s} be a set of s + 1 matrices in PAut(Hm ). Then, Ps is an s-PD-set of size s + 1 for Hm with information set Im if and only if no two matrices (Mi−1 )∗ and (Mj−1 )∗ for i 6= j have a row in common. Moreover, any subset Pk ⊆ Ps of size k + 1 is a k-PD-set for k ∈ {1, . . . , s}. Proof. Suppose that the set Ps = {Mi : 0 ≤ i ≤ s} satisfies that no two matrices (Mi−1 )∗ and (Mj−1 )∗ for i 6= j have a row in common. Let E = {v1 , . . . , vs } ⊆ {1} × Zm 2 be a set of s different column vectors of the generator matrix Gm regarded as row vectors, which represents a set of s error positions. Assume we cannot move all the error positions to the check set by any element of Ps . Then, for each i ∈ {0, . . . , s}, there is a v ∈ E such that vMi ∈ Im . In other words, there is at least an error position that remains in the information set Im after applying any permutation of Ps . Note that there are s + 1 values for i, but only s elements in E. Therefore, vMi ∈ Im and vMj ∈ Im for some v ∈ E and i 6= j. Suppose vMi = wr and vMj = wt , for wr , wt ∈ Im . Then, v = wr Mi−1 = wt Mj−1 . Taking into account the form of the vectors in the information set Im = {w1 , . . . , wm+1 }, by multiplying for such inverse matrices Mi−1 and Mj−1 , we get the first row or a certain addition between the first row and another row of each matrix. Thus, we obtain that (Mi−1 )∗ and (Mj−1 )∗ have a row in common, contradicting our assumption. Let Pk ⊆ Ps 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. 5 Conversely, suppose that the set Ps = {Mi : 0 ≤ i ≤ s} forms an s-PDset for Hm , but does not satisfy the condition on the inverse matrices. Thus, some v ∈ {w1 , . . . , w2m } must be the rth row of (Mi−1 )∗ and the tth row of (Mj−1 )∗ for some r, t ∈ {1, . . . , m + 1}, i, j ∈ {0, . . . , s}. In other words, we have that v = er (Mi−1 )∗ = et (Mj−1 )∗ . Therefore, v = wr Mi−1 = wt Mj−1 , where wr , wt ∈ Im . Finally, we obtain that vMi = wr and vMj = wt . These equalities implies that the vector v, which represents an error position, cannot be moved to the check set by the permutations defined by matrices Mi and Mj . Let L = {l : 0 ≤ l ≤ s, l 6= i, j}. For each l ∈ L, choose a row vl of (Ml−1 )∗ . It is clear that vl = et (Ml−1 )∗ = wt Ml−1 , so vl Ml = wt ∈ Im . Finally, since some of the vl may repeat, we obtain a set E = {vl : l ∈ L} ∪ {v} of size at most s. Nevertheless, no matrix in Ps will map every member of E into the check set, fact that contradicts our assumption. We give now an explicit construction of an fm -PD-set {M0 , . . . , Mfm } ⊆ PAut(Hm ) of minimum size fm +1 for the binary linear Hadamard code Hm of length 2m . We follow a similar technique to the one described for simplex codes in [4]. Lemma 6. Let K = Z2 [x]/(f (x)), where f (x) is a primitive polynomial of degree m. If α is a root of f (x), then αi+1 − αi , . . . , αi+m − αi are linearly independent over Z2 , for all i ∈ {0, . . . , 2m − 2}. Proof. It is straightforward to see that αi+1 − αi , . . . , αi+m − αi are linearly independent over Z2 , for all i ∈ {0, . . . , 2m − 2}, if and only if α − 1, . . . , αm − 1 are linearly independent over Z2 , since αi ∈ K\{0}. Pm−1 Note that αm − 1 = j=1 µj αj , where the sum has and odd number of nonzero terms, since f (x) is irreducible. Let µ = (µ1 , . . . , µm−1 ) ∈ Z2m−1 . Note that in vectorial notation αj − 1 = e1 + ej , j ∈ {1, . . . , m − 1} and Pm−1 αm − 1 = j=1 µj ej+1 . Finally, it is easy to see that the m × m binary matrix   1 Idm−1 , 0 µ Pm−1 which has as rows α − 1, . . . , αm − 1, has determinant j=1 µj = 1 6= 0. For i ∈ {1, . . . , fm }, consider the following (m+1)×(m+1) binary matrices:     1 α(m+1)i−1 1 0  0  0  1  α(m+1)i − α(m+1)i−1     N0 =  . and N =   . .. .. .. i  ..    . . . 0 αm−1 0 α(m+1)i+m−1 − α(m+1)i−1 Theorem 7. Let Pfm = {Mi : 0 ≤ i ≤ fm }, where Mi = Ni−1 . Then, Pfm is an fm -PD-set of size fm + 1 for the binary linear Hadamard code Hm of length 2m with information set Im . Proof. Clearly, N0 ∈ PAut(Hm ), since it is the identity matrix. By Lemma 6, Ni ∈ PAut(Hm ) for all i ∈ {1, . . . , fm }. Moreover, rows of matrices N0∗ , . . . , Nf∗m form the set {(1, a) : a ∈ {0, 1, α, . . . , αfm (m+1)+m−1 }}. The elements of such set are different since α is primitive and fm (m + 1) + m − 1 ≤ 2m − 2. Theorem 5 completes the proof. 6 Example 8. Let H4 be the binary linear Hadamard code of length 16 with generator matrix (4). Let K = Z2 [x]/(x4 + x + 1) and α a root of x4 + x + 1. Matrices N0 = Id5 ,        1 1 1 0 0 1 α4 1 α9  0 α10 − α9    0 α5 − α4   0 1 0 1 0         6 4      0 α11 − α9  N1 =  0 α − α  =  0 1 1 1 1  , N2 =  =   0 α7 − α4   0 0 0 0 1   0 α12 − α9   0 α8 − α4 0 0 1 1 0 0 α13 − α9 where Id5 is the 5 × 5 identity matrix, are elements of PAut(H4 ) and P2 = {N0−1 , N1−1 , N2−1 } is a 2-PD-set of size 3 for H4 . It is straightforward to chech that matrices N0∗ ,     1 1 1 0 0 1 0 1 0 1  1 0 1 1 0   1 1 1 1 0      ∗ ∗    N1 =  1 0 0 1 1  , and N2 =   1 0 1 1 1 ,  1 1 1 0 1   1 1 1 1 1  1 1 0 1 0 1 1 0 1 1 have no rows in common. Finally, no s-PD-set of size s + 1 can be found for s ≥ 3 since f4 = 2. Let S be an s-PD-set of size s+1. The set S is a nested s-PD-set if there is an ordering of the elements of S, S = {σ0 , . . . , σs }, such that Si = {σ0 , . . . , σi } ⊆ S is an i-PD-set of size i + 1, for all i ∈ {0, . . . , s}. Note that Si ⊂ Sj if 0 ≤ i < j ≤ s and Ss = S. From Theorem 5, we have two important consequences. The first one is related to how to obtain nested s-PD-sets and the second one provides another proof of Lemma 3. Corollary 9. Let m be an integer, m ≥ 4. If Ps is an s-PD-set of size s + 1 for the binary linear Hadamard code Hm , then any ordering of the elements of Ps gives nested k-PD-sets for k ∈ {1, . . . , s}. Corollary 10. Let m be an integer, m ≥ 4. If Ps is s-PD-set of size s + 1 k j an m −m−1 . for the binary linear Hadamard code Hm , then s ≤ 2 1+m Proof. Following the condition on sets of matrices to be s-PD-sets of size s + 1, given by Theorem 5, we have to obtain certain s + 1 matrices with no rows in common. Note that the number of possible vectors of length m + 1 over Z2 with 1 in the first coordinate is 2m . Thus, taking this fact into account and counting the number of rows of each one of these s +j1 matrices, k we have that 2m −m−1 2m m . (s + 1)(m + 1) ≤ 2 , so s + 1 ≤ m+1 and finally s ≤ 1+m 4 Recursive construction of s-PD-sets for binary linear Hadamard codes In this section, given an s-PD-set of size l for the binary linear Hadamard code Hm of length 2m , where l ≥ s + 1, we show how to construct recursively an ′ s-PD-set of the same size for Hm′ of length 2m for all m′ > m. 7 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0    ,   Given a matrix M ∈ PAut(Hm ) and an integer κ ≥ 1, we define the matrix M (κ) ∈ PAut(Hm+κ ) as   M 0 M (κ) = , (5) 0 Idκ where Idκ denotes the κ × κ identity matrix. Proposition 11. Let m be an integer, m ≥ 4, and Ps = {Mi : 0 ≤ i ≤ s} be an s-PD-set of size s + 1 for Hm with information set Im . Then, Qs = {(Mi−1 (κ))−1 : 0 ≤ i ≤ s} is an s-PD-set of size s + 1 for Hm+κ with information set Im+κ , for any κ ≥ 1. Proof. Since Ps is an s-PD-set for Hm , matrices (M1−1 )∗ , . . . , (Ms−1 )∗ have no rows in common by Theorem 5. Therefore, it is straightforward to check that matrices (M1−1 (κ))∗ , . . . , (Ms−1 (κ))∗ have no rows in common either. Moreover, Mi−1 (κ) ∈ PAut(Hm+κ ), for all i ∈ {1, . . . , s}. Thus, applying again Theorem 5, we have that Qs is an s-PD-set for Hm+κ . It is important to note that the bound fm+1 for Hm+1 cannot be achieved recursively from an s-PD-set for Hm , since the above recursive construction works for a given fixed s, increasing the length of the Hadamard code. The above recursive construction only holds when the size of the s-PD-set is exactly s + 1. Now, we will show a second recursive construction which holds when the size of the s-PD-set is any integer l, l ≥ s+1. In this case, the elements of PAut(Hm ) will be regarded as permutations of coordinate positions, that is, as elements of Sym(2m ) instead of matrices of GL(m + 1, 2). It is well known that a generator matrix Gm+1 for the binary linear Hadamard code Hm+1 of length 2m+1 can be constructed as follows:   Gm Gm , (6) Gm+1 = 0 1 where Gm is a generator matrix for the binary linear Hadamard code Hm of length 2m . Given two permutations σ1 ∈ Sym(n1 ) and σ2 ∈ Sym(n2 ), we define (σ1 |σ2 ) ∈ Sym(n1 + n2 ), where σ1 acts on the coordinates {1, . . . , n1 } and σ2 on {n1 + 1, . . . , n1 + n2 }. Proposition 12. Let m be an integer, m ≥ 4, and S be an s-PD-set of size l for Hm with information set I. Then, (S|S) = {(σ|σ) : σ ∈ S} is an sPD-set of size l for Hm+1 constructed from (6), with any information set I ′ = I ∪ {i + 2m }, i ∈ I. Proof. Since I is an information set for Hm , we have that |(Hm )I | = 2m+1 . Since Hm+1 is constructed from (6), it follows that Hm+1 = {(x, x), (x, x̄) : x ∈ Hm }, where x̄ is the complementary vector of x. A vector and its complementary have different values in each coordinate, so |(Hm+1 )I∪{i} )| = 2m+2 , for all i ∈ {2m + 1, . . . , 2m+1 }. Thus, any set of the form I ′ = I ∪ {i + 2m }, i ∈ I, is an information set for Hm+1 . If σ ∈ PAut(Hm ), then σ(x) = z ∈ Hm for all x ∈ Hm . Therefore, since (σ|σ)(x, x) = (z, z) and (σ|σ)(x, x̄) = (z, z + σ(1)) = (z, z̄), we can conclude that (σ|σ) ∈ PAut(Hm+1 ). 8 n Let e = (a, b) ∈ Z2n 2 , where a = (a1 , . . . , an ), b = (b1 , . . . , bn ) ∈ Z2 , and n = 2m . Finally, we will prove that for every e ∈ Z2n with wt(e) ≤ s, there is 2 (σ|σ) ∈ (S|S) such that (σ|σ)(e)I ′ = 0. Let c = (c1 , . . . , cn ) be the binary vector defined as follows: ci = 1 if and only if ai = 1 or bi = 1, for all i ∈ {1, . . . , n}. Note that wt(c) ≤ s, since wt(e) ≤ s. Taking into account that S is an s-PD-set with respect to I, there is σ ∈ S such that σ(c)I = 0. Therefore, we also have that (σ|σ)(a, b)I∪J = 0, where J = {i + n : i ∈ I}. The result follows trivially since I ′ ⊆ I ∪ J. 5 Finding s-PD-sets of size s + 1 for Hadamard Z4 -linear codes   }, there is an unique (up to equivaFor any m ≥ 3 and each δ ∈ {1, . . . , m+1 2 lence) Z4 -linear Hadamard code of length 2m which is the Gray map image of a quaternary linear code of length β = 2m−1 and type 2γ 4δ , where m = γ +2δ −1. Moreover, for a fixed m, all these codes are pairwise nonequivalent, except for δ = 1 and δ = 2, since these ones are equivalent to the binary linear Hadamard code of length 2m [10]. Therefore,  the number of nonequivalent  for all m ≥ 3. Note that when Z4 -linear Hadamard codes of length 2m is m−1 2 δ ≥ 3, the Z4 -linear Hadamard codes are nonlinear. Let Hγ,δ be the quaternary linear Hadamard code of length β = 2m−1 and type 2γ 4δ , where m = γ + 2δ − 1, and let Hγ,δ = Φ(Hγ,δ ) be the corresponding Z4 -linear code of length 2β = 2m . A generator matrix Gγ,δ for the code Hγ,δ can be constructed by using the following recursive constructions:   Gγ,δ Gγ,δ , (7) Gγ+1,δ = 0 2   Gγ,δ Gγ,δ Gγ,δ Gγ,δ , (8) Gγ,δ+1 = 0 1 2 3 starting from G0,1 = (1). We first obtain G0,δ from G0,1 by using recursively δ times construction (8). Then, Gγ,δ is managed from G0,δ by using γ times construction (7). Note that the rows of order four remain in the upper part of Gγ,δ while those of order two stay in the lower part. A set I = {i1 , . . . , iγ+δ } ⊆ {1, . . . , β} of γ + δ coordinate positions is said to be a quaternary information set for a quaternary linear code C of type 2γ 4δ if |CI | = 2γ 4δ . If the coordinates in I are ordered in such a way that |C{i1 ,...,iδ } | = 4δ , it is easy to see that the set Φ(I), defined as Φ(I) = {2i1 − 1, 2i1 , . . . , 2iδ − 1, 2iδ , 2iδ+1 − 1, . . . , 2iδ+γ − 1}, is an information set for C = Φ(C). For example, the set I = {1} is a quaternary information set for H0,1 , so Φ(I) = {1, 2} is an information set for H0,1 = Φ(H0,1 ). In general, there is not an unique way to obtain a quaternary information set for the code Hγ,δ . The following result provides a recursive and simple form to obtain such a set. Proposition 13. Let I be a quaternary information set for the quaternary linear Hadamard code Hγ,δ of length β = 2m−1 and type 2γ 4δ , where m = γ +2δ −1. Then I ∪{β +1} is a quaternary information set for the codes Hγ+1,δ and Hγ,δ+1 , which are obtained from Hγ,δ by applying (7) and (8), respectively. 9 Proof. Since |Hγ+1,δ | = 2γ+1 4δ and |Hγ,δ+1 | = 2γ 4δ+1 , it is clear that a quaternary information set for codes Hγ+1,δ and Hγ,δ+1 should have γ + δ + 1 = |I| + 1 coordinate positions. Taking into account that Hγ,δ+1 is constructed from (8), we have that Hγ,δ+1 = {(u, u, u, u), (u, u+1, u+2, u+3), (u, u+2, u, u+2), (u, u+3, u+2, u+ 1) : u ∈ Hγ,δ }. Vectors u, u+1, u+2, and u+3 have different values in each coordinate, so |(Hγ,δ+1 )I∪{i} | = 2γ 4δ+1 for all i ∈ {β + 1, . . . , 2β, 3β + 1, . . . , 4β}. In particular, I ∪ {β + 1} is a quaternary information set for Hγ,δ+1 . A similar argument holds for Hγ+1,δ . Since Hγ+1,δ is constructed from (7), we have that Hγ+1,δ = {(u, u), (u, u + 2) : u ∈ Hγ,δ }. Vectors u and u + 2 have different values in each coordinate, so |(Hγ+1,δ )I∪{i} | = 2γ+1 4δ for all i ∈ {β + 1, . . . , 2β}. Therefore, I ∪ {β + 1} is a quaternary information set for Hγ+1,δ . Despite the fact that the quaternary information set I ∪ {β + 1}, given by Proposition 13, is the same for Hγ+1,δ and Hγ,δ+1 , the information set for the corresponding binary codes Hγ+1,δ and Hγ,δ+1 are I ′ = Φ(I) ∪ {2β + 1} and I ′′ = Φ(I) ∪ {2β + 1, 2β + 2}, respectively. As for binary linear codes, we can label the ith coordinate position of a quaternary linear code C, with the ith column of a generator matrix G of C. Thus, any quaternary information set I for C can also be considered as a set of vectors representing the positions in I. Then, by Proposition 13, we have that the set Iγ,δ = {e1 , e1 + e2 , . . . , e1 + eδ , e1 + 2eδ+1 , . . . , e1 + 2eγ+δ } is a suitable quaternary information set for the code Hγ,δ . Depending on the context, Iγ,δ will be considered as a subset of {1, . . . , β} or as a subset of {1} × Z4δ−1 × {0, 2}γ . Example 14. The quaternary generated by the matrix  1 1 1 1 1 G0,3 =  0 1 2 3 0 0 0 0 0 1 linear Hadamard code H0,3 of length 16 can be  1 1 1 1 1 1 1 1 1 1 1 1 2 3 0 1 2 3 0 1 2 3 , 1 1 1 2 2 2 2 3 3 3 3 obtained by applying two times construction (8) over G0,1 = (1). The set I0,3 = {1, 2, 5}, or equivalently the set of column vectors I0,3 = {(1, 0, 0), (1, 1, 0), (1, 0, 1)} of G0,3 , is a quaternary information set for H0,3 . By applying constructions (7) and (8) over G0,3 , we obtain that matrices   G0,3 G0,3 , G1,3 = 0 2   G0,3 G0,3 G0,3 G0,3 , G0,4 = 0 1 2 3 generate the quaternary linear Hadamard codes H1,3 and H0,4 of length 32 and 64, respectively. By Propositions 13, it follows that I0,3 ∪ {17} = {1, 2, 5, 17} is a quaternary information set for H1,3 and H0,4 . Despite the fact that the quaternary information set is the same for both codes H1,3 and H0,4 , it is important to note that in terms of column vectors representing these positions, we have that I1,3 = {(1, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 2)} and I0,4 = {(1, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1)}. Finally, I ′ = Φ(I0,3 ) ∪ {33} = {1, 2, 3, 4, 9, 10, 33} and I ′′ = Φ(I0,3 ) ∪ {33, 34} = {1, 2, 3, 4, 9, 10, 33, 34} are information sets for the Z4 -linear Hadamard codes H1,3 and H0,4 , respectively. 10 Let C be a quaternary linear code of length β and type 2γ 4δ , and let C = Φ(C) be the corresponding Z4 -linear code of length 2β. Let Φ : Sym(β) → Sym(2β) be the map defined as  2τ (i/2), if i is even, Φ(τ )(i) = 2τ ((i + 1)/2) − 1 if i is odd, for all τ ∈ Sym(β) and i ∈ {1, . . . , 2β}. Given a subset S ⊆ Sym(β), we define the set Φ(S) = {Φ(τ ) : τ ∈ S} ⊆ Sym(2β). It is easy to see that if S ⊆ PAut(C) ⊆ Sym(β), then Φ(S) ⊆ PAut(C) ⊆ Sym(2β). Let GL(k, Z4 ) denote the general linear group of degree k over Z4 and let L be the set consisting of all matrices over Z4 of the following form:   1 η 2θ  0 A 2X  , 0 Y B where A ∈ GL(δ −1, Z4 ), B ∈ GL(γ, Z4 ), X is a matrix over Z4 of size (δ −1)×γ, Y is a matrix over Z4 of size γ × (δ − 1), η ∈ Z4δ−1 and θ ∈ Zγ4 . Lemma 15. The set L is a subgroup of GL(γ + δ, Z4 ). Proof. We first need to check that L ⊆ GL(γ + δ, Z4 ), in other words, that det(M) ∈ {1, 3} (that is, an unit of Z4 ) for all M ∈ L. Note that if M′ ∈ GL(k, Z4 ), then M = M′ + 2R ∈ GL(k, Z4 ). Thus, since det(M′ ) ∈ {1, 3}, we have that det(M) ∈ {1, 3}, where   1 η 0 M′ =  0 A 0  . 0 Y B It is straightforward to check that MN ∈ L for all M, N ∈ L. Let ζ be the map from Z4 to Z4 which is the usual modulo two map composed with inclusion from Z2 to Z4 , that is ζ(0) = ζ(2) = 0, ζ(1) = ζ(3) = 1. This map can be extended to matrices over Z4 by applying ζ to each one of their entries. Let π be the map from L to L defined as   1 η 2θ A 2X  , π(M) =  0 0 ζ(Y ) ζ(B) and let π(L) = {π(M) : M ∈ L} ⊆ GL(γ + δ, Z4 ). By Lemma 15, it is clear that π(L) is a group with the operation ∗ defined as M ∗ N = π(MN ) for all M, N ∈ π(L). By the proof of Theorem 2 in [9], it is easy to see that the permutation automorphism group PAut(Hγ,δ ) of Hγ,δ is isomorphic to π(L). Thus, from now on, we identify PAut(Hγ,δ ) with this group. Recall that we can label the ith coordinate position of Hγ,δ with the ith column vector wi of the generator matrix Gγ,δ constructed via (7) and (8), i ∈ {1, . . . , β}. Therefore, again, any matrix M ∈ PAut(Hγ,δ ) can be seen as a permutation of coordinate positions τ ∈ Sym(β), such that τ (i) = j as long as wj = wi M, i, j ∈ {1, . . . , β}. For any M ∈ PAut(Hγ,δ ), we define Φ(M) = Φ(τ ) ∈ Sym(2β), and for any P ⊆ PAut(Hγ,δ ), we consider Φ(P) = {Φ(M) : M ∈ P} ⊆ Sym(2β). 11 Lemma 16. Let Hγ,δ be the quaternary linear Hadamard code of length β and type 2γ 4δ and let P ⊆ PAut(Hγ,δ ). Then, Φ(P) is an s-PD-set for Hγ,δ with information set Φ(Iγ,δ ) if and only if for every s-set E of column vectors of Gγ,δ there is M ∈ P such that {gM : g ∈ E} ∩ Iγ,δ = ∅. Proof. If Φ(P) is an s-PD-set with respect to the information set Φ(Iγ,δ ), then for every s-set E ⊆ {1, . . . , 2β}, there is τ ∈ P ⊆ Sym(β) such that Φ(τ )(E) ∩ Φ(Iγ,δ ) = ∅. For every s-set E ⊆ {1, . . . , β}, let Eo = {2i − 1 : i ∈ E}. We know that there is τ ∈ P such that Φ(τ )(Eo ) ∩ Φ(Iγ,δ ) = ∅. By the definition of Φ, we also have that τ (E) ∩ Iγ,δ = ∅, which is equivalent to the statement. Conversely, we assume that for every s-set E ⊆ {1, . . . , β}, there is τ ∈ P ⊆ Sym(β) such that τ (E) ∩ Iγ,δ = ∅. For every s-set E ⊆ {1, . . . , 2β}, let Eo be an s-set such that {i : ϕ1 (i) ∈ E or ϕ2 (i) ∈ E} ⊆ Eo , where ϕ1 (i) = 2i − 1 and ϕ2 (i) = 2i. Since there is τ ∈ P such that τ (Eo ) ∩ Iγ,δ = ∅, we have that Φ(τ )(E) ∩ Φ(Iγ,δ ) = ∅. Let M ∈ PAut(Hγ,δ ) and let mi be the ith row of M, i ∈ {1, . . . , δ + γ}. We define M∗ as the matrix where the first row is m1 and the ith row is m1 + mi for i ∈ {2, . . . , δ} and m1 + 2mi for i ∈ {δ + 1, . . . , δ + γ}. Theorem 17. Let Hγ,δ be the quaternary linear Hadamard code of type 2γ 4δ . Let Ps = {Mi : 0 ≤ i ≤ s} be a set of s + 1 matrices in PAut(Hγ,δ ). Then, Φ(Ps ) is an s-PD-set of size s + 1 for Hγ,δ with information set Φ(Iγ,δ ) if and −1 ∗ ∗ only if no two matrices (M−1 i ) and (Mj ) for i 6= j have a row in common. Proof. By Lemma 16 and following a similar argument as in the proof of Theorem 5. Corollary 18. Let Ps be a set of s + 1 matrices in PAut(Hγ,δ ). If Φ(Ps ) is an s-PD-set of size s + 1 for Hγ,δ , then any ordering of elements in Φ(Ps ) provides nested k-PD-sets for k ∈ {1, . . . , s}. Corollary 19. Let Ps be a set of s + 1 matrices in PAut(Hγ,δ ). If Φ(Ps ) is an s-PD-set of size s + 1 for Hγ,δ , then s ≤ fγ,δ , where fγ,δ =   2γ+2δ−2 − γ − δ . γ+δ Proof. Following the condition on sets of matrices to be s-PD-sets of size s + 1, given by Theorem 17, we have to obtain certain s + 1 matrices with no rows in common. Since the rows of length δ + γ must have 1 in the first coordinate, and elements from {0, 2} in the last γ coordinates, the number of possible rows is 4δ−1 2γ = 2γ+2δ−2 . Thus, taking this fact into account and counting the number of rows of each one of these s+1 matrices, we have that (s+1)(γ +δ) ≤ 2γ+2δ−2 , and the result follows. We give now an explicit construction of an f0,δ -PD-set of size f0,δ + 1 for H0,δ . Let R = GR(4δ−1 ) be the Galois extension of dimension δ − 1 over Z4 . It is known that R is isomorphic to Z4 [x]/(h(x)), where h(x) is a monic basic irreducible polynomial of degree δ − 1. Let f (x) ∈ Z2 [x] be a primitive polynomial of degree δ − 1. Let ℓ = 2δ−1 − 1. There is a unique primitive basic irreducible polynomial h(x) dividing xℓ − 1 in Z4 [x]. Let T = {0, 1, α, . . . , αℓ−1 } ⊆ R, 12 where α is a root of h(x). It is well known that any r ∈ R can be written uniquely as r = a + 2b, where a, b ∈ T . We take R as the following ordered set: R= = {r1 , . . . , r4δ−1 } {0 + 2 · 0, . . . , αℓ−1 + 2 · 0, . . . , 0 + 2 · αℓ−1 , . . . , αℓ−1 + 2 · αℓ−1 }. Since |R|/δ = f0,δ + 1, we can form f0,δ + 1 disjoints sets of R of size δ. For all i ∈ {0, . . . , f0,δ }, we consider the δ × δ quaternary matrix   1 rδi+1   .. Ni∗ =  ... . . 1 rδ(i+1) Theorem 20. Let Pf0,δ = {Mi : 0 ≤ i ≤ f0,δ }, where Mi = Ni−1 . Then, Φ(Pf0,δ ) is an f0,δ -PD-set of size f0,δ + 1 for the Z4 -linear Hadamard code H0,δ of length 22δ−1 . Proof. We need to prove that rδi+2 − rδi+1 , . . . , rδ(i+1) − rδi+1 are linearly independent over Z4 , for all i ∈ {0, . . . , f0,δ }, to guarantee that Ni ∈ PAut(H0,δ ). Note that these vectors are not zero divisors [7]. Since αℓ = 1, {rδi+2 − rδi+1 , . . . , rδ(i+1) − rδi+1 } is one of the following three sets: L1 = L2 = L3 = {1, . . . , αδ−2 }, {αk+1 − αk , . . . , αk+δ−1 − αk }, for some k ∈ {0, . . . , ℓ − 1}, {αk+1 − αk , . . . , αℓ−1 − αk , −αk + 2(bj − bi ), αℓ − αk + 2(bj − bi ), . . . , αk+δ−2 − αk + 2(bj − bi )}, for some bi , bj ∈ T and k ∈ {0, . . . , ℓ − 1}. Elements in L1 are clearly linearly independent over Z4 . Now we prove that the same property is satisfied in L2 . Assume the contrary that there are some P onk+i λi 6= 0, i ∈ {1, . . . , δ − 1}, such that λi (α − αk ) = 0. If λi ∈ {1, 3} for at least one i ∈ {1, . . . , δ − 1}, we get a contradiction. P Indeed, if we take modulo 2 in the previous linear combination, we obtain that λ̄i (ᾱk+i − ᾱk ) = 0, where λ̄i ∈ Z2 and at least one λ̄i 6= 0. This is a contradiction by Lemma 6. On the other hand, ifPλi ∈ {0, 2} for all i ∈P{1, . . . , δ − 1} and there is at least one λi = 2, then 2λ′i (αk+i − αk ) =P2[ λ′i (αk+i − αk )] = 0, where λ′i ∈ {0, 1} λ′i (αk+i − αk ) = 2λ for some λ ∈ R, that and at least one λ′i = 1. Hence, is, it is a zero divisor. By taking modulo 2, we obtain a contradiction again by Lemma 6. We show that elements in L3 = {v1 , . . . , vδ−1 } are also linearly independent over Z4 by using a slight modification of the previous argument. Suppose that P there is at least one λi 6= 0, i ∈ {1, P . . . , δ − 1}, such that λi v i P = 0. By taking modulo 2, we obtain that λ̄δ−1 ᾱk + λ̄i (ᾱk+i −P ᾱk ) = ᾱk [λ̄δ−1 + λ̄i (ᾱi −1)] = k 0. Since ᾱ is a unit, it follows that λ̄δ−1 + λ̄i (ᾱi − 1) = 0, which gives a contradiction if λi ∈ {1, 3} for at least one index, since 1, ᾱ − 1, . . . , ᾱδ−2 − 1 are linearly independent over Z2 . Note that the binary matrix   1 0 , 1 Idm−1 has determinant 1. If λi ∈ {0, 2} for all i ∈ {1, . . . , δ − 1}, we get a contradiction by applying a similar argument to the one used above. Finally, by construction, matrices Ni∗ have no rows in common, since their rows are different elements of R. By Theorem 17, the result follows. 13 Example 21. Let H0,3 be the quaternary linear Hadamard code of length 16 and type 20 43 . Let R = Z4 [x]/(h(x)), where h(x) = x2 + x + 1. Note that h(x) is a primitive basic irreducible polynomial dividing x3 − 1 in Z4 [x]. Let α be a root of h(x). Then, T = {0, 1, α, α2 } and elements in R are ordered as follows: R= = It is easy to  1 N1∗ =  1 1 {r1 , . . . , r16 } {0, 1, α, 3 + 3α, 2, 3, 2 + α, 1 + 3α, 2α, 1 + 2α, 3α, 3 + α, 2 + 2α, 3 + 2α, 2 + 3α, 1 + α}. check that matrices N0∗ = Id∗3 ,    1 2 1 3 3 2 0 , N2∗ =  1 1 3  , 3 0 1 0 2  1 1 2 N3∗ =  1 0 3  , 1 3 1  have no rows in common. Let P4 = {N0−1 , N1−1 , N2−1 , N3−1 , N4−1 }, Id3 ,      1 1 1 2 1 1 3 3 N1 =  0 3 1  , N3 =  0 3 N2 =  0 3 2  , 0 2 0 2 1 0 0 1   1 2 2 N4∗ =  1 3 2  , 1 2 3 where N0 =  2 1 , 3  1 2 2 N4 =  0 1 0  . 0 0 1 The set Φ(P4 ) is a 4-PD-set of size 5 for H0,3 . Note that the bound f5 = 4 is attained for H0,3 despite the search of the 4-PD-set is done in the subgroup Φ(PAut(H0,3 )) ≤ PAut(H0,3 ), since f0,3 = f5 = 4. We have that the Z4 -linear Hadamard code of length 2m with δ = 1 or δ = 2 is equivalent to the binary linear Hadamard code of length 2m [10]. However, the technique explained for binary linear Hadamard codes in Section 3 provides better results (in terms of s) that the one explained for Z4 -linear Hadamard codes when applied for linear codes, since fγ,δ ≤ fm , where m = γ + 2δ − 1. Example 22. We have provided a 2-PD-set of size 3 for the binary linear Hadamard code H4 of length 16 in Example 8. The code H4 is equivalent to both Z4 -linear Hadamard codes H1,2 and H3,1 . However, a 2-PD-set of size 3 is not achievable for H4 by using Theorem 17, since f1,2 = f3,1 = 1. Example 23. The binary linear Hadamard code H5 of length 32 admits a 4PD-set of size 5 by Theorem 5, since f5 = 4. Considering H5 as the Gray map image of the quaternary linear Hadamard code H2,2 or H4,1 , no more than a 3-PD-set of size 4 can be found by using Theorem 17, since f4,1 = 2 and f2,2 = 3. 6 Recursive construction of s-PD-sets for Z4 -linear Hadamard codes In this section, given an s-PD-set of size l for the Z4 -linear Hadamard code Hγ,δ of length 2m and type 2γ 4δ , where m = γ + 2δ − 1 and l ≥ s + 1, we show how to construct recursively an s-PD-set of the same size for Hγ+i,δ+j of length 2m+i+2j and type 2γ+i 4δ+j for all i, j ≥ 0. 14  We first provide a recursive construction considering the elements of PAut(Hγ,δ ) as matrices in GL(γ +δ, Z4 ). This construction can be seen as a natural generalization of the technique introduced for binary linear Hadamard codes in Section 4. Given a matrix M ∈ PAut(Hγ,δ ) and an integer κ ≥ 1, we define   1 η 0 2θ  0 A 0 2X  . M(κ) =  (9)  0 0 Idκ 0  0 ζ(Y ) 0 ζ(B) Proposition 24. Let Ps = {M0 , . . . , Ms } ⊆ PAut(Hγ,δ ) such that Φ(Ps ) is an s-PD-set of size s + 1 for Hγ,δ with information set Φ(Iγ,δ ). Then, Qs = −1 −1 {(M−1 , . . . , M−1 } ⊆ PAut(Hγ+i,δ+j ) and Φ(Qs ) is an s-PD-set s (κ)) 0 (κ)) of size s + 1 for Hγ+i,δ+j with information set Φ(Iγ+i,δ+j ), for any i, j ≥ 0 such that i + j = κ ≥ 1. Proof. Note that if M ∈ PAut(Hγ,δ ), construction (9) provides an element M(κ) ∈ GL(γ + δ + κ, Z4 ). Taking this into account, together with the fact that Idκ can split as   Idj 0 Idκ = , 0 Idi where i + j = κ ≥ 1, it is obvious that M−1 (κ) ∈ PAut(Hγ+i,δ+j ) and so its inverse. Thus, Qs ⊆ PAut(Hγ+i,δ+j ). Finally, repeated rows in matrices ∗ −1 ∗ (M−1 0 (κ)) , . . . , (Ms (κ)) cannot occur, since this fact implies repeated rows −1 ∗ ∗ in matrices (M0 ) , . . . , (M−1 s ) by construction (9). The result follows from Theorem 17. Example 25. Let P4 = {M0 , . . . , M4 } ⊆ PAut(H0,3 ) be the set, given in Example 21, such that Φ(P4 ) is a 4-PD-set of size 5 for H0,3 . By Proposition −1 24, the set Q4 = {M−1 : 0 ≤ i ≤ 4} is contained in both PAut(H1,3 ) i (1)) and PAut(H0,4 ). Moreover, Φ(Q4 ) is a 4-PD-set of size 5 for H1,3 and H0,4 . ∗ Nevertheless, it is important to note that the construction of (M−1 i (1)) depends −1 on the group where Mi (1) is considered. As for binary linear Hadamard codes, a second recursive construction considering the elements of PAut(Hγ,δ ) as permutations of coordinate positions, that is as elements of Sym(2m ), can also be provided. Given four permutations σi ∈ Sym(ni ), i ∈ {1, . . . , 4}, we define (σ1 |σ2 |σ3 |σ4 ) ∈ Sym(n1 + n2 + n3 + n4 ) in the same way as we defined (σ1 |σ2 ) ∈ Sym(n1 + n2 ) in Section 4. Proposition 26. Let S be an s-PD-set of size l for Hγ,δ of length n and type 2γ 4δ with information set I. Then, (S|S) = {(σ|σ) : σ ∈ S} is an s-PD-set of size l for Hγ+1,δ of length 2n and type 2γ+1 4δ constructed from (7) and the Gray map, with any information set I ′ = I ∪ {i + n}, i ∈ I. Proof. Since Hγ+1,δ = {(x, x), (x, x̄) : x ∈ Hγ,δ }, where x̄ is the complementary vector of x, the result follows using the same argument as in the proof of Proposition 12. By the proof of Proposition 13, we can add any of the coordinate positions of {i + n : i ∈ I} to I in order to form a suitable information set I ′ for Hγ+1,δ . 15 Proposition 26 cannot be generalized directly for Z4 -linear Hadamard codes Hγ,δ+1 constructed from (8) and the Gray map. Note that if S is an s-PDset for Hγ,δ , then (S|S|S|S) = {(σ|σ|σ|σ) : σ ∈ S} is not always an s-PDset for Hγ,δ+1 , since in general (σ|σ|σ|σ) ∈ / PAut(Hγ,δ ). For example, σ = (1, 5)(2, 8, 3, 6, 4, 7) ∈ PAut(H0,2 ) ⊆ Sym(8), but π = (σ|σ|σ|σ) ∈ / PAut(H0,3 ) ⊆ Sym(32), since π(Φ((0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3))) = Φ((0, 0, 0, 0, 0, 2, 0, 2, 2, 2, 2, 2, 2, 0, 2, 0)) 6∈ H0,3 . Proposition 27. Let S ⊆ PAut(Hγ,δ ) such that Φ(S) is an s-PD-set of size l for Hγ,δ of length n and type 2γ 4δ with information set I. Then, Φ((S|S|S|S)) = {Φ((τ |τ |τ |τ )) : τ ∈ S} is an s-PD-set of size l for Hγ,δ+1 of length 4n and type 2γ 4δ+1 constructed from (8) and the Gray map, with any information set I ′′ = I ∪ {i + n, j + n}, i, j ∈ I and i 6= j. Proof. Since Hγ,δ+1 is constructed from (8), Hγ,δ+1 = {(u, u, u, u), (u, u + 1, u + 2, u + 3), (u, u + 2, u, u + 2), (u, u + 3, u, u + 1) : u ∈ Hγ,δ }. It is easy to see that if τ ∈ PAut(Hγ,δ ), then (τ |τ |τ |τ ) ∈ PAut(Hγ,δ+1 ). Let σ = Φ(τ ). Finally, we need to prove that for every e ∈ Z4n 2 with wt(e) ≤ s, there is (σ|σ|σ|σ) ∈ Φ((S|S|S|S)) such that (σ|σ|σ|σ)(e)I ′′ = 0, where I ′′ ⊆ {1, . . . , 4n} is an information set for Hγ,δ+1 with γ+2(δ+1) coordinate positions. Using a similar argument to that given in the proofs of Propositions 12 and 26, the result follows. Moreover, by the proof of Proposition 13, any I ′′ = I ∪ {i + n, j + n} with i, j ∈ I and i 6= j is a suitable information set for Hγ,δ+1 . Propositions 26 and 27 can be applied recursively to acquire s-PD-sets for any Z4 -linear Hadamard codes obtained (by using constructions (7) and (8)) from a given Z4 -linear Hadamard code where we already have such set. With this aim in mind, let denote by 2S the set (S|S) and by 2i S = 2(2(.i). .(2S)). Corollary 28. Let S ⊆ PAut(Hγ,δ ) such that Φ(S) is an s-PD-set of size l for Hγ,δ of length 2m and type 2γ 4δ with information set I. Then, Φ(2i+2j S) is an s-PD-set of size l for Hγ+i,δ+j of length 2m+i+2j and type 2γ+i 4δ+j with information set obtained by applying recursively Proposition 13, for all i, j ≥ 0. Proof. The result comes trivially by applying Propositions 13, 26 and 27. Note that, from Theorem 20 and Proposition 26, we have explicitly provided an f0,δ -PD-set of size f0,δ +1 for each nonlinear Z4 -linear Hadamard code Hγ,δ , γ ≥ 0, δ ≥ 3. 7 Conclusions An alternative permutation decoding method that can be applied to Z2 Z4 linear codes [3], which include Z4 -linear codes, was presented in [2]. However, it remained as an open question to determine PD-sets for some families of Z2 Z4 linear codes. In this paper, the problem of finding s-PD-sets of minimum size s+1 for binary linear and Z4 -linear Hadamard codes is addressed by finding s+1 invertible matrices over Z2 or Z4 , respectively, which satisfy certain conditions. Moreover, note that the first examples and constructions of (nonlinear) Z4 -linear Hadamard codes are provided. This approach establishes equivalent results to the ones obtained for simplex codes in [4]. 16 As a future research in this topic, it would be interesting to provide explicitly an fγ,δ -PD-set of size fγ,δ + 1 for Hγ,δ , s-PD-sets of minimum size for Z2 Z4 linear Hadamard codes in general [13], or other families of Z4 -linear codes such as Kerdock codes [7]. References [1] R. Barrolleta and M. Villanueva, “Partial permutation decoding for binary linear Hadamard codes,” Electron. Note Discr. Math. 46, 35–42 (2014). [2] J. J. Bernal, J. Borges, C. Fernández-Córboda, and M. Villanueva, “Permutation decoding of Z2 Z4 -linear codes,” Des. Codes and Cryptogr. 76(2), 269–277 (2015). [3] J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifà, and M. Villanueva, “Z2 Z4 -linear codes: generator matrices and duality,” Des. Codes and Cryptogr. 54, 167–179 (2010). [4] W. Fish, J. D. 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