Butson full propelinear codes

In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of \'{O} Cath\'{a}in and Swartz. That is, we show how, if given a Butson Hadamard matrix over the $k^{\rm th}$ roots of unity, we can construct a larger Butson matrix over the $\ell^{\rm th}$ roots of unity for any $\ell$ dividing $k$, provided that any prime $p$ dividing $k$ also divides $\ell$. We prove that a $\mathbb{Z}_{p^s}$-additive code with $p$ a prime number is isomorphic as a group to a BH-code over $\mathbb{Z}_{p^s}$ and the image of this BH-code under the Gray map is a BH-code over $\mathbb{Z}_p$ (binary Hadamard code for $p=2$). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided.


Introduction
Let n and k be positive integers, and ζ k = exp (2π √ −1/k) be the complex k th root of unity.We write ζ k = {ζ j k } 0≤j≤k−1 .Let Z k be the ring of integers modulo k with k > 1, and denote by Z n k the set of n-tuples over Z k .We use bold notation x = [x 1 , . . ., x n ] ∈ Z n k to denote vectors (or codewords) in Z n k .We denote the set of n × n matrices with entries in a set X by M n (X).

Butson Hadamard matrices
A Butson Hadamard (or simply Butson) matrix of order n and phase k is a matrix H ∈ M n ( ζ k ) such that HH * = nI n , where I n denotes the identity matrix of order n and H * denotes the conjugate transpose of H.We write BH(n, k) for the set of such matrices.The simplest examples of Butson matrices are the Fourier matrices ] n i,j=1 ∈ BH(n, n).Hadamard matrices of order n, as they are usually defined, are the elements of BH(n, 2).The phase and orthogonality of a BH(n, k) is preserved by multiplication on the left or right by a n × n monomial matrix with non-zero entries in the k th roots of unity.For any pair of such monomial matrices (P, Q) the operation defined by H(P, Q) = P HQ * = H ′ is an equivalence operation, and H and H ′ are said to be equivalent.If H = H ′ , then (P, Q) is an automorphism of H.
A Butson matrix H ∈ BH(n, k) is conveniently represented in logarithmic form, that is, the matrix H = [ζ ϕ i,j k ] n i,j=1 is represented by the matrix L(H) = [ϕ i,j mod k] n i,j=1 with the convention that L i,j ∈ Z k for all i, j ∈ {1, . . ., n}.Observe that the matrix above is in dephased form, that is, its first row and column are all 0. Every matrix can be dephased by using equivalence operations.Throughout this paper all matrices are assumed to be dephased.
Example 1.2.Let p be a prime number.If L(D) = [xy T ] x,y∈Z n p then D is a BH(p n , p).In fact D is the n-fold Kronecker product of the Fourier matrix of order p.When p = 2 this is the well known Sylvester Hadamard matrix of order 2 n .
Butson matrices have been subject to a considerable increase in interest recently for a variety of reasons.For one, a BH(n, k) exists for all n, (the Fourier matrix for example), but real Hadamard matrices, i.e., BH(n, 2), exist when n > 2 only if n ≡ 0 mod 4, and this condition is famously not yet known to be sufficient.A Butson morphism [8] is a map BH(n, k) → BH(m, ℓ).This motives the study of Butson matrices even if real Hadamard matrices are the primary interest.In Section 2.2 we construct a morphism BH(n, k) → BH(nm, k/m) where k = p e 1 1 • • • p et t and m = p e 1 −1 , matching the parameters of the morphism discovered by Ó Catháin and Swartz in [18].But their applications in applied sciences most strongly motivate their study.A BH(n, k) scaled by a factor of 1/ √ n is an orthonormal basis of C n .In any set of mutually unbiased bases (MUBs) which includes the standard basis, all other bases are necessarily of this form.MUBs have important applications in quantum physics, such as yielding optimal schemes of orthogonal quantum measurement (see e.g., [2]).Butson matrices also have applications in coding theory, as we discuss next.
For a code C ⊆ Z n k , we denote by Aut(C) the group of all isometries of Z n k fixing the code C and we call it the automorphism group of the code C. A code C over Z k is called transitive if Aut(C) acts transitively on its codewords, i.e., the code satisfies the property (i) of the above definition.
Assuming that C has a propelinear structure then a binary operation ⋆ can be defined as x ⋆ y = (σ x , π x )(y) for any x, y ∈ C. Therefore, (C, ⋆) is a group, which is not abelian in general.This group structure is compatible with the Hamming distance, that is, The vector 0 is always a codeword where π 0 = Id n is the identity coordinate permutation and σ 0,i = Id k is the identity permutation on Z k for all i ∈ {1, . . ., n}.Hence, 0 is the identity element in C and π x −1 = π −1 x and σ x −1 ,i = σ −1 x,πx(i) for all x ∈ C and for all i ∈ {1, . . ., n}.We call (C, ⋆) a propelinear code.Henceforth we use C instead of (C, ⋆) if there is no confusion.Definition 1.5.A full propelinear code is a propelinear code C such that for every a ∈ C, σ a (x) = a + x and π a has not any fixed coordinate when a = α1 for α ∈ Z k .Otherwise, π a = Id n .
Remark 1.6.Every linear code is propelinear but not necessarily full.
A Butson Hadamard code, which is also full propelinear, is called a Butson Hadamard full propelinear code (briefly, BHFP-code).In the binary case, we have the Hadamard full propelinear codes, they were introduced in [23] and their equivalence with Hadamard groups was proven.In the q-ary case, i.e., codes over the finite field F q where q is a prime power, the generalized Hadamard full propelinear codes were introduced in [1].Their existence is shown to be equivalent to the existence of central relative (n, q, n, n/q).
Propelinear codes are a topic of increasing interest in algebraic coding theory.The primary reason for this is that they offer one of the main benefits of linear codes, which is that they can be entirely described by a few generating codewords and group relations.However as the codes are not necessarily linear, they are not subject to all of the same minimum distance constraints as linear codes with the same number of codewords.Some propelinear codes may outperform comparable linear codes by having a larger minimum distance that any linear code of the same size, or by having more codewords than any linear code with a given minimum distance [1,11].In this paper we extend the work of the authors in [1] and describe the connection between cocyclic Butson Hadamard matrices and BHFP-codes.

Constructing Butson Hadamard matrices and related codes
Throughout this paper we study BH-codes over Z k .We have already introduced the Lee and Hamming distance between vectors x and y.We define other useful distance functions here.Initially, let k = p s for a prime p.The weight function wt * (x) with x ∈ Z p s is defined by For p = s = 2, this is the Lee weight.The corresponding distance d * on Z n p s is defined as follows: where The definition of the weight function here is consistent with the homogeneous metric introduced in [5].The corresponding distance d † on Z n mp s is defined as follows: where [16] gives a pattern that any row of L(H) has to follow.That is, any row x has to be a permutation of the vector As a consequence, n − n p is an upper bound for the minimum Hamming distance of F H when k = p s .Furthermore, the minimum Hamming distance of both codes, F H and C H , is the same in this case.
In [19,21], the authors prove that if n = p sm and k = p s then the minimum Hamming distance of F H is n − n p and the minimum Lee distance is given by where H is the Butson matrix of Theorem 2.3 and m = t 1 − 1 for t 1 > 0 and t 2 = . . .= t s = 0. Finally, Theorem 5.4 of [10] claims that for any pair (n, k) such that BH(n, k) = ∅, if H ∈ BH(n, k) then the code obtained by deleting the first coordinate in F H has parameters (n − 1, n, γn) meeting the Plotkin bound over Frobenius rings where γ is the average homogeneous weight over Z k .

A Fourier type construction and simplex codes
In what follows, we describe a method to construct Butson matrices of order n = p st 1 +(s−1)t 2 +...+ts−s and phase k = p s , where p is a prime.Let s be a positive integer, t 1 , t 2 , . . ., t s be nonnegative integers with t 1 ≥ 1, and A 1,0,...,0 = [0].The matrix A t 1 ,t 2 ,...,ts , where p i−1 denotes the all-p i−1 vector, is defined recursively according to the following algorithm, where initially, By construction, it is clear that A t 1 ,t 2 ,...,ts is a (t 1 + t 2 + . . .+ t s ) × (p st 1 +(s−1)t 2 +...+ts−s ) matrix.This is a generalization of the construction of [19] as we will point out in Corollary 2.8.
Proof.By construction, the difference between two distinct rows of L(H) is a linear combination (with coefficients in Z p s ) of the rows of A t 1 ,t 2 ,...,ts .Hence, it is a row of L(H).
Therefore, proving HH * = nI n reduces to proving that every row sum of H is 0. For the rows of H corresponding to multiples of the rows of A t 1 ,t 2 ,...,ts , this holds as a consequence of Lemma 2.2.Finally, the proof for the rows of H corresponding to a linear combination of the rows of A t 1 ,t 2 ,...,ts is by a simple induction.
We provide some examples of Butson matrices coming from Theorem 2.3.(8,8) where  .In general we have the following.Proposition 2.6.Let n = p st 1 +(s−1)t 2 +...+ts−s and L(H) be the n × n matrix of Theorem 2.3.Then, H is equivalent to where F p s−j denotes the Fourier matrix of order p s−j embedded in BH(p s−j , p s ) using that ζ p s−j = ζ p j p s , and (M) r denotes the r-fold Kronecker product of the matrix M.
. For the next step of the induction, we assume that t i+1 = . . .= t s = 0 and . Now, we have to distinguish two possibilities: Proceeding in a similar way, the result holds.
It is clear now that this construction is not new, in the sense that it does not produce any Butson matrices not already known.However this perspective gives us new insights into the related BH-codes.
Corollary 2.8.A simplex code of type α over Z 2 s of length 2 sm (see [19]) and the code whose codewords are the rows of L((F 2 s ) m ) are the same.Therefore M ψ , the cocyclic BH(2 sm , 2 s ) of [19, Theorem 5.1, ii)] is equivalent to (F 2 s ) m .Similarly, when p > 2 prime, the analogous classifying result for the cocyclic BH(p sm , p s ) of [21, Proposition 3.1, ii)] holds.
Proof.Attending to the Remark above, a simplex code of type α over Z 2 s of length n = 2 st 1 is exactly the code F H where H is the n × n matrix of Theorem 2.3.Applying Proposition 2.6, the results follows.
The classifying result above follows also as a consequence of [17,Theorem 13].
A nonempty subset C of Z n p s is a Z p s -additive code if it is a subgroup of Z n p s (i.e., a Z p s -module).Clearly, given a Z p s -additive code, C, of length n there exist some non negative integers t 1 , . . ., t s such that C is isomorphic (as an abelian group) to Z t 1 p s × Z t 2 p s−1 × . . .× Z ts p .Thus, C is said to be of type (n; t 1 , . . ., t s ).Note that |C| = p st 1 p (s−1)t 2 . . .p ts since there are t 1 (generators) codewords of order p s , t 2 of order p s−1 and so on.
Proposition 2.10.For t 1 > 0, every H t 1 ,...,ts is a BH-code where the Butson Hadamard matrix is a Kronecker product of Fourier matrices.
Proof.Let C H be the BH-code associated to H of Theorem 2.3.It is clear that C H is equivalent to H t 1 ,...,ts .Now, the result follows from Proposition 2.6.The following is an example of a BH-code which is not additive.
Example 2.11.Let H ∈ BH (8,4) with additive since the double of the second row is not a codeword.Now, we can state that, in a certain sense, the class of BH-codes encompasses strictly the class of Z p s -additive codes.Since any Z p s -additive code is always of type (n; t 1 , . . ., t s ) for some non negatives integers t 1 , . . ., t s and H t 1 ,...,ts ∈ BH(p st 1 +(s−1)t 2 +...+ts−s , p s ), assuming that t 1 > 0.

Generalized Gray map
The Gray map is a function from Z 4 to Z 2  2 which is typically used to form binary codes from Z 4 -codes.In what follows, we introduce a generalized Gray map Φ p from Z p s to Z p s−1 p , and extend this to a yet more general function Let us observe that for p = 2, Φ p is the well-known Carlet's map [4] and for p > 2, Φ p is of type ϕ given in [24].For what remains of this section we write Φ = Φ p for brevity unless there is some confusion.
Further, by the linearity of the inner product vw T and the definition .
Then H Φ is the corresponding matrix in M np s−1 ( ζ p ).

Proof.
Observe that H Φ is Butson Hadamard over ζ p if, for all i = j, the sequence of differences [L(H Φ )] i,l − [L(H Φ )] j,l , 0 ≤ l ≤ n • p s−1 − 1 contains each element of Z p equally often.First note that for all i = j, the sequence of differences [L(H)] i,l − [L(H)] j,l , 0 ≤ l ≤ n − 1 contains each element of the form ap s−1 equally often for a = 0, . . ., p − 1.This is a consequence of ζ ap s−1 k being a p th root of unity.By Lemma 2.13, if x − y = ap s−1 then Φ(x − y) = Φ(x) − Φ(y).Since Φ(ap s−1 ) = a1 for a ∈ Z p , it follows that if the set of differences [L(H)] i,l − [L(H)] j,l contains m repetitions of each element of the form ap s−1 , then the set of corresponding differences in   Now let k = mp s where p does not divide m and recall that every element x ∈ Z k can be written uniquely as x = ap s + bm mod k for some 0 ≤ a ≤ m − 1 and 0 ≤ b ≤ p s − 1.Then let Proposition 2.18.The entrywise application of Ψ p is an isometric embedding of (Z n mp s , d † ) into (Z p s−1 n mp , d H ). Furthermore, if C is a code with parameters (n, M, d † ) over Z mp s , then the image code C = Ψ p (C) is a code with parameters (p s−1 n, M, d H ) over Z mp .
Proof.This follows from a straight forward extension of Proposition 2.12. .
Then H Ψp is the corresponding matrix in M np s−1 ( ζ pm ).We will devote the rest of this section to a proof of the following.Before we can prove Theorem 2.19, we will need to establish some preliminary results.Hereafter we fix a prime p and let Ψ = Ψ p .i=1 ω Ψ(z) i = 0 where ω is a primitive k th root of unity.Otherwise, Ψ(z) = f 1, and Corollary 2.23.If x = f p s−1 and y = 0 mod p s−1 , then Ψ(x − y) = Ψ(x) − Ψ(y) + m(p − 1)1.Consequently, for any multiset X of elements of Z k such that x ∈ X only if x = f p s−1 ,and for any y = 0 mod p s−1 , then We will require the following result of Lam and Leung.Lemma 2.24 (Corollary 3.2, [15]).If α 1 +• • •+α r = 0 is a minimal vanishing sum of n th roots of unity, then after a suitable rotation, we may assume that all α i 's are n th 0 roots of unity where n 0 is square-free.
Proposition 2.27.Let C be a BH-code of minimum Hamming distance d obtained from a BH(n, p s m) with p a prime not dividing m.Then the minimum distance Remark 2.28.The upper bound above is attainable.For example, the code C obtained from the Fourier matrix of order 27 has minimum distance 18.The code Ψ(C) is a BH-code of length 243, with minimum distance 162 = 18(3 2 ).

Propelinear codes and cocyclic matrices
The BH-matrix given in Example 2.11, H, is cocyclic over Z 8 and its BHcode associated C H is not linear.Can we define a propelinear structure in C H ? Certainly, we can and this is not an isolated situation.
Let G and U be finite groups, with U abelian, of orders n and k, respectively.A map ψ : is a cocycle (over G, with coefficients in U).We may assume that ψ is normalized, i.e., ψ(g, 1) = ψ(1, g) = 1 for all g ∈ G.For any (normalized) map φ : G → U, the cocycle ∂φ defined by ∂φ(g, h) = φ(g) −1 φ(h) −1 φ(gh) is a coboundary.The set of all cocycles ψ : G × G → U forms an abelian group Z 2 (G, U) under pointwise multiplication.Factoring out the subgroup of coboundaries gives H 2 (G, U), the second cohomology group of G with coefficients in U.
Given a group G and ψ ∈ Z 2 (G, U), denote by E ψ the canonical central extension of U by G; this has elements {(u, g) | u ∈ U, g ∈ G} and multiplication (u, g) (v, h) = (uvψ(g, h), gh).The image U × {1} of U lies in the centre of E ψ and the set ) is displayed as a cocyclic matrix M ψ : under some indexing of the rows and columns by G, M ψ has entry ψ(g, h) in position (g, h).
A n × n matrix A = (a g,h ) g,h∈G is called G-invariant (or just group invariant) if a gk,hk = a g,h for all g, h, k ∈ G.
Remark 3.2.Every group invariant matrix with entries in U is equivalent to a cocyclic matrix.
Fact: A cocyclic Butson Hadamard matrix is not necessarily pairwise row and column balanced. .From part 1 we know that x + π x (y) is a scalar multiple of the n-tuple defined by ψ(hg, −), and thus the j th component of π x+πx(y) (z) is ψ(ℓ, hgg j ).Now observe that the k th component of π y (z) is ψ(ℓ, hg k ).We have π x (k) = j where g k = gg j , and thus the j th component of π x (π y (z)) is ψ(ℓ, hg k ) = ψ(ℓ, hgg j ).Proof.Extend the definition of π x for the rows x of L(H ψ ) to all of C H by letting π x+α1 = π x for all α ∈ Z k .The code C H is propelinear by Proposition 3.4, and since x ⋆ y = x + π x (y) for all x, y ∈ C, the first property of Definition 1.5 is satisfied.Finally observe that because π x ∈ S n is defined so that π −1 x (j) = k where g k = gg j , it follows that π x fixes no coordinate when x = α1, and π α1 = Id Sn for all α ∈ Z k .Remark 3.6.A notorious class of cocyclic Butson matrices are those that are equivalent to group invariant (if G is a cyclic group, they are called circulant Butson matrices).A construction method based on bilinear forms on finite abelian groups is given in [6] which, in turn, provides BHFP-codes.Furthermore, for G abelian it is known that Bent functions, group invariant generalized Hadamard matrices and abelian semiregular relative different sets are all either equivalent to group invariant Butson matrices or to group invariant Butson matrices with additional properties (see [25]).Characterising group invariant Butson matrices in terms of BHFP codes is an open problem.
We refer the reader to [1, Section 3] for a detailed discussion on cocyclic generalized Hadamard matrices and the corresponding generalized Hadamard full propelinear codes.Rather than repeat this discussion, we note that the converse of Corollary 3.5 holds under the assumption that the BH(n, k) is row and column balanced.A BH(n, p) is necessarily balanced, and is equivalent to a generalized Hadamard matrix over the cyclic group C p when p is prime.
can be endowed with a full propelinear structure with the following group Π of permutations x ∈ C 5 (1,6,3,8,5,2,7,4) x ∈ C 6 (1, 7, 5, 3) (2,8,6,4) x ∈ C 7 (1,8,7,6,5,4,3,2) x ∈ C 8 C H is a BHFP-code with group structure Z 8 ×Z 4 and Π ∼ = Z 8 .The codewords are An interesting family of BH-codes over Z p s are those associated to Kronecker products of Fourier matrices.They are denoted by H t 1 ,t 2 ,...,ts (see Remark 2.9 and Proposition 2.10) and since these matrices are cocyclic over p s , these codes can be endowed with a full propelinear structure by Corollary 3.5 .Furthermore, for p = 2 and s = 2 in [20], it is shown that the image of H t 1 ,t 2 under the Gray map are in fact propelinear codes.The full propelinear code is a group

Propelinear codes via the Gray map
A natural question that arises is whether or not the generalized Gray preserves the property of being propelinear, or full propelinear.It is certainly true that the number of codewords in a BH-code C obtained from H, a BH(n, mp s ), is the same as the number of codewords in the BH-code C ′ obtained from H Ψ .However, in general, it is not the case that C ′ will be an isomorphic propelinear structure.A simple example to demonstrate this arises from the Z 9 -code C obtained from the trivial BH(1, 9), and the Z 3code Ψ(C) obtained from the BH(3, 3) matrix H ′ = (1) Ψ which written in log form is The code C is clearly linear, and as a group is isomorphic to the cyclic group Z 9 .It is also easily seen to be full propelinear by definition.However it is a short exercise to verify that Ψ(C) cannot be both full propelinear and isomorphic to a cyclic group G ∼ = Z 9 generated by any single element x, no matter what the coordinate permutation π x may be.The code Ψ(C) does form a 2-dimensional linear code (so it is also propelinear, but not full propelinear with x ⋆ y = x + y for all x, y ∈ Ψ(C)), and Ψ is a bijective map between codewords, but in general it is not always the case that Ψ(x ⋆ y) = Ψ(x) ⋆ ′ Ψ(y) for any operation ⋆ ′ , and as a consequence Ψ will generally not preserve a group structure.The code Ψ(C) of this example can also be with a full propelinear structure, but it will not be isomorphic as a group to C. It is generated by the codewords x = [0, 1, 2], and 1, where π x = (1, 3, 2).It is isomorphic to Z 2 3 .However, we find that for the special case Ψ 2 : Z 4m → Z 2 2m , we can carefully construct an isomorphism between the groups of codewords C and C ′ = Ψ 2 (C), and determine the group operation ⋆ ′ so that (C, ⋆) ∼ = (C ′ , ⋆ ′ ).Let Ψ = Ψ 2 .Proof.First observe that Ψ is a bijection from C to C ′ , so we need to determine the group of permutations for C ′ and show that Ψ : (C, ⋆) → (C ′ , ⋆ ′ ) is a homomorphism.We start with the n = 1 case, so we just need to show that we can choose ρ x ∈ S 2 for each x ∈ Z 4m so that Ψ(x) + ρ x (Ψ(y)) = Ψ(x + y) for all y.We will see that ρ x = (1, 2) x , i.e., ρ x permutes the two coordinates of a word in Z 2 2m or not, according to the parity of x.We adhere to the notation of the proof of Lemma 2.21.Fix x = 4a + mb and let y = 4c + md where 0 ≤ b, d ≤ 3, so x+y = 4(a+c)+m(b+d) with the value of b+d taken modulo 4. A complete proof requires a verification that Ψ(x) + ρ x (Ψ(y)) = Ψ(x + y) for each pair (b, d) ∈ Z 4 , but for brevity we take (b, d) = (3, 1) as an example The corresponding permutations ρ x π Φ(x) and ρ y π Φ(y) are as follows: ρ x π Φ(x) = (1, 7, 6, 4)(2, 8, 5, 3)(9, 15, 14, 12) (10,16,13,11) (17,23,22,20) (18,24,21,19) (25,31,30,28) (26,32,29,27), ρ y π Φ(y) = (1, 25, 17, 9)(2, 26, 18, 10) (3,28,19,12) (4,27,20,11) (5, 29, 21, 13)(6, 30, 22, 14) (7,32,23,16) (8,31,24,15).

Example 1 . 1 .
he following is a BH(4, 8) matrix H, display in logarithmic form

1 p
this function we construct a morphism BH(n, k) → BH(nk/ℓ, ℓ).Where x = [x 1 , . . ., x n ] ∈ Z n k and ϕ is any function with domain Z k , we will write ϕ(x) = [ϕ(x 1 ), . . ., ϕ(x n )].Further, we write ϕ(C) = {ϕ(c) : c ∈ C} where C ⊆ Z n k .We consider the elements of Z s−1 p to be ordered in increasing lexicographic order.We denote by D the BH(p s−1 , p) matrix defined in Example 1.2 and label the rows of L(D) in the order 0, 1, . . ., p s−1 − 1.Let [L(D)] i denotes the row of L(D) labeled by i.Then we let Φ p : Z p s → Z p s−be the map defined by Φ p (x) = [L(D)] b + a1, x = ap s−1 + b.

Corollary 3 . 7 .
Let C H be a BHFP-code of length n over Z k coming from H ∈ BH(n, k), where H is row and column balanced.Then H is cocyclic.Proof.The proof follows the proof of Proposition 4 and Corollary 2 of[1].Let H be a BH(n, k).We consider the following partition of its corresponding code.C H = ∪ 1≤α≤n C α where C α = {[L(H)] α +λ1} λ∈Z k and [L(D)] i denotes the i-th row of L(D).Example 3.8.Let H be the BH-matrix of Example 2.11 since it is cocyclic over Z 8 .Then,

Theorem 4 . 1 .
Let m be an odd positive integer, and let C ⊆ Z n 4m be a full propelinear code.Then the code C ′ = Ψ(C) is full propelinear with group structure (C ′ , ⋆ ′ ) ∼ = (C, ⋆).
where we have used that F 2 ⊗ F 4 ∈ BH(8, 8) by means of ζ 2 = ζ 4 8 and ζ 4 = ζ 2 8 [7,5.The image of any BH-code over Z p s of length n by Φ is a BH-code over Z p of length n • p s−1 and minimum Hamming distance d H = np s−2 (p − 1).Remark 2.16.Let us point out that Theorem 1 of[9]is a particular case of Corollary 2.15 (when the BH-code is of type H t 1 ,...,ts and p = 2).Any BH-code C H of length n over Z p s has minimum distance d * = np s−2 (p − 1).Proof.Taking into account that BH(n, p) = GH(p, n/p) where GH(p, n/p) denotes the set of generalized Hadamard matrices of order n over F p (see[7,  Lemma 2.2]).Thus, C H Φ = Φ(C H ) is a generalized Hadamard code as well since H Φ ∈ BH(p s−1 n, p).The minimum Hamming distance of these codes is well known to be np s−2 (p − 1).The fact that Φ is an isometric embedding (Proposition 2.12) concludes the proof.