Butson full propelinear codes José Andrés Armario∗ Universidad de Sevilla, Sevilla, Spain arXiv:2010.06206v1 [cs.IT] 13 Oct 2020 Ivan Bailera† Universitat Autònoma de Barcelona, Bellaterra, Spain Ronan Egan‡ National University of Ireland Galway, Galway, Ireland October 14, 2020 Abstract 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 Ó Catháin and Swartz. That is, we show how, if given a Butson Hadamard matrix over the kth roots of unity, we can construct a larger Butson matrix over the ℓth roots of unity for any ℓ dividing k, provided that any prime p dividing k also divides ℓ. We prove that a Zps -additive code with p a prime number is isomorphic as a group to a BH-code over Zps and the image of this BH-code under the Gray map is a BH-code over Zp (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. Keywords: Cocycles, Butson Hadamard matrices, Gray map, propelinear codes. Mathematics Subject Classification (2010): 05B20, 05E18, 94B60. ∗ E-mail:armario@us.es E-mail:ivan.bailera@uab.cat ‡ E-mail:ronan.egan@nuigalway.ie † 1 1 Introduction √ Let n and k be positive integers, and ζk = exp (2π −1/k) be the complex k th root of unity. We write hζk i = {ζkj }0≤j≤k−1. Let Zk be the ring of integers modulo k with k > 1, and denote by Znk the set of n-tuples over Zk . We use bold notation x = [x1 , . . . , xn ] ∈ Znk to denote vectors (or codewords) in Znk . We denote the set of n × n matrices with entries in a set X by Mn (X). 1.1 Butson Hadamard matrices A Butson Hadamard (or simply Butson) matrix of order n and phase k is a matrix H ∈ Mn (hζk i) such that HH ∗ = nIn , where In 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 ma(i−1)(j−1) n trices are the Fourier matrices Fn = [ζn ]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 ϕ form, that is, the matrix H = [ζk i,j ]ni,j=1 is represented by the matrix L(H) = [ϕi,j mod k]ni,j=1 with the convention that Li,j ∈ Zk for all i, j ∈ {1, . . . , n}. Example 1.1. he following is a BH(4, 8) form  0 0  0 2 L(H) =   0 4 0 6 matrix H, display in logarithmic  0 0 4 6   0 4  4 2 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∈Znp then D is a BH(pn , 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 2n . 2 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 = pe11 · · · pet t and m = pe11 −1 · · · pet t −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 Cn . 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. 1.2 BH-codes and propelinear codes Interest in studying codes over finite rings increased significantly after it was proved in [11] that certain notorious non-linear binary codes (such as the Preparata codes or the Kerdock codes), which had some of the properties of linear codes were, in fact, the images of codes over Z4 under a non-linear map (the Gray map). Codes constructed from Butson matrices [10, 17, 19, 21] are a particular type of codes over a finite ring. A code over Zk (or Zk code) of length n is a nonempty subset C of Znk . The elements of C are called codewords. The Hamming weight of a vector x ∈ Zk , denoted by wtH (x), is the number of nonzero coordinates of x. The Hamming distance between two vectors x, y ∈ Znk , denoted by dH (x, y) = wtH (x − y), is the number of coordinates in which they differ. Given a minimum Hamming distance d = minx,y∈C,x6=y dH (x, y) for a code C of length n, we say C is a (n, |C|, d) code. Other distances functions are used, for instance, the Lee distance between two vectors x, y ∈ Znk is dL (x, y) = wtLP (x − y) where the n Lee weight of a vector z = [z1 , . . . , zn ] ∈ Zk is wtL (z) = ni=1 wtL (zi ) with wtL (zi ) = min{zi , k − zi }. Given H ∈ BH(n, k), we denote by FH the Zk -code of length n consisting of the rows of L(H), and by CH the Zk -code defined as CH = ∪α∈Zk (FH +α1) where 1 denotes the all-one vector (and α1 the all-α vector). We will write 1n to denote the all-one vector of length n when clarification is required. The code CH over Zk is called a Butson Hadamard code (briefly, BH-code). 3 Example 1.3. Given H ∈ BH(4, 8) of Example 1.1. Then FH = {[0, 0, 0, 0], [0, 2, 4, 6], [0, 4, 0, 4], [0, 6, 4, 2]},  [0, 0, 0, 0], [0, 2, 4, 6], [0, 4, 0, 4], [0, 6, 4, 2],     [1, 1, 1, 1], [1, 3, 5, 7], [1, 5, 1, 5], [1, 7, 5, 3],     [2, 2, 2, 2], [2, 4, 6, 0], [2, 6, 2, 6], [2, 0, 6, 4],    [3, 3, 3, 3], [3, 5, 7, 1], [3, 7, 3, 7], [3, 1, 7, 5], CH = [4, 4, 4, 4], [4, 6, 0, 2], [4, 0, 4, 0], [4, 2, 0, 6],     [5, 5, 5, 5], [5, 7, 1, 3], [5, 1, 5, 1], [5, 3, 1, 7],     [6, 6, 6, 6], [6, 0, 2, 4], [6, 2, 6, 2], [6, 4, 2, 0],    [7, 7, 7, 7], [7, 1, 3, 5], [7, 3, 7, 3], [7, 5, 3, 1]             .            Assuming the Hamming metric, any isometry of Znk is given by a coordinate permutation π and n permutations σ1 , . . . , σn of Zk . We denote by Aut(Znk ) the group of all isometries of Znk : Aut(Znk ) = {(σ, π) : σ = (σ1 , . . . , σn ) with σi ∈ Sym Zk , π ∈ Sn } where Sym Zk and Sn denote, respectively, the symmetric group of permutations on Zk and on the set {1, . . . , n}. The action of (σ, π) is defined as (σ, π)(v) = σ(π(v)) for any v ∈ Znk , and the group operation in Aut(Zk ) is the composition (σ, π) ◦ (σ ′ , π ′ ) = ((σ1 ◦ σπ′ −1 (1) , . . . , σn ◦ σπ′ −1 (n) ), π ◦ π ′ ) for all (σ, π), (σ ′ , π ′ ) ∈ Aut(Zk ). Definition 1.4. A code C of length n over Zk has a propelinear structure if for any codeword x ∈ C there exist πx ∈ Sn and σx = (σx,1 , . . . , σx,n ) with σx,i ∈ Sym Zk satisfying: (i) (σx , πx )(C) = C and (σx , πx )(0) = x, (ii) if y ∈ C and z = (σx , πx )(y), then (σz , πz ) = (σx , πx ) ◦ (σy , πy ). The propelinear structure was introduced in [22] for binary codes, and it was generalized in [3] for q-ary codes. For a code C ⊆ Znk , we denote by Aut(C) the group of all isometries of Znk fixing the code C and we call it the automorphism group of the code C. A code C over Zk is called transitive if Aut(C) acts transitively on its codewords, i.e., the code satisfies the property (i) of the above definition. 4 Assuming that C has a propelinear structure then a binary operation ⋆ can be defined as x ⋆ y = (σx , πx )(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, dH (x ⋆ u, x ⋆ v) = dH (u, v) where u, v ∈ Znk . The vector 0 is always a codeword where π0 = Idn is the identity coordinate permutation and σ0,i = Idk is the identity permutation on Zk for all i ∈ {1, . . . , n}. Hence, 0 is the identity element in −1 C and πx−1 = πx−1 and σx−1 ,i = σx,π for all x ∈ C and for all i ∈ {1, . . . , n}. x (i) 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, σa (x) = a + x and πa has not any fixed coordinate when a 6= α1 for α ∈ Zk . Otherwise, πa = Idn . 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 Fq 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. 5 2 Constructing Butson Hadamard matrices and related codes Throughout this paper we study BH-codes over Zk . We have already introduced the Lee and Hamming distance between vectors x and y. We define other useful distance functions here. Initially, let k = ps for a prime p. The weight function wt∗ (x) with x ∈ Zps is defined by   (p − 1)ps−2 x 6= kps−1 mod ps , k ∈ Zp ps−1 x = kps−1 mod ps , k ∈ Zp \ {0} wt∗ (x) =  0 x = 0 mod ps For p = s = 2, this is the Lee weight. The corresponding distance d∗ on Znps is defined as follows: d∗ (x, y) = n X i=1 wt∗ (yi − xi ), (1) where x = [x1 , . . . , xn ] and y = [y1 , . . . , yn ] in Znps . More generally, let k = mps for m coprime to p. Any x ∈ Zk may be written uniquely in the form x = aps + bm mod k where 0 ≤ a ≤ m − 1 and 0 ≤ b ≤ ps − 1. Define the weight function wt† (x) on Zk by  wt∗ (b) a = 0 † wt (x) = ps−1 a 6= 0. The definition of the weight function here is consistent with the homogeneous metric introduced in [5]. The corresponding distance d† on Znmps is defined as follows: n X † d (x, y) = wt† (yi − xi ), (2) i=1 where x = [x1 , . . . , xn ] and y = [y1 , . . . , yn ] in Znmps . Given H ∈ BH(n, k), recall that FH is the Zk -code of length n consisting i of the rows of L(H), and CH = ∪α∈Zk (FH + α1). Let rH (l) be the number i of repetitions of l ∈ Zk in the i-th row of L(H) and rH (l) = max rH (l). 2≤i≤n For k = ps , Lemma 3.1 of [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 (u, r1 1 + u, . . . , rt−1 1 + u) where u = [0, ps−1, 2ps−1 , . . . , , (p − 1)ps−1], ri ∈ Zps−1 , for 1 ≤ i ≤ t − 1 with t = np . Therefore,  n l = hps−1 where h ∈ Zp p rH (l) ≤ n − 1 Otherwise. p 6 As a consequence, n − np is an upper bound for the minimum Hamming distance of FH when k = ps . Furthermore, the minimum Hamming distance of both codes, FH and CH , is the same in this case. In [19, 21], the authors prove that if n = psm and k = ps then the minimum Hamming distance of FH is n − np and the minimum Lee distance is given by  2m+s−2 , p=2 dL = s(m+1)−2 p 2 (p − 1), p > 2 prime; 4 where H is the Butson matrix of Theorem 2.3 and m = t1 − 1 for t1 > 0 and t2 = . . . = ts = 0. Finally, Theorem 5.4 of [10] claims that for any pair (n, k) such that BH(n, k) 6= ∅, if H ∈ BH(n, k) then the code obtained by deleting the first coordinate in FH has parameters (n − 1, n, γn) meeting the Plotkin bound over Frobenius rings where γ is the average homogeneous weight over Zk . 2.1 A Fourier type construction and simplex codes In what follows, we describe a method to construct Butson matrices of order n = pst1 +(s−1)t2 +...+ts −s and phase k = ps , where p is a prime. Let s be a positive integer, t1 , t2 , . . . , ts be nonnegative integers with t1 ≥ 1, and A1,0,...,0 = [0]. The matrix At1 ,t2 ,...,ts , where pi−1 denotes the all-pi−1 vector, is defined recursively according to the following algorithm, where initially, (t′1 , t′2 , . . . , t′s ) = (1, 0, . . . , 0). for i=1 until s do while t′i < ti do ′ ′ A ← At1 ,...,ts t′i ← t′i + 1  ′ ′ At1 ,...,ts ← Ai = end while end for A A ... A 0 · pi−1 1 · pi−1 . . . (ps−i+1 − 1) · pi−1  By construction, it is clear that At1 ,t2 ,...,ts is a (t1 + t2 + . . . + ts ) × (pst1 +(s−1)t2 +...+ts −s ) matrix. This is a generalization of the construction of [19] as we will point out in Corollary 2.8.   0 0 0 0 1,1,0 Example 2.1. For p = 2 and s = 3. We have A = and 0 2 4 6 7 A1,1,1   0 0 0 0 0 0 0 0 =  0 2 4 6 0 2 4 6 . 0 0 0 0 4 4 4 4 Given x ∈ Znk , the order of x is the smallest positive integer m such that mx = 0 over Zk . Lemma 2.2. Let k = ps and ui = [ 0 · pi−1 , 1 · pi−1 , . . . , (ps−i+1 − 1) · pi−1 ] ∈ s−i+1 where 1 ≤ i ≤ s. Then Zpps ps−i+1 −1 • X j=0 ζkjp i−1 = 0, for all 1 ≤ i ≤ s. • The order of ui is ps−i+1 . • If gcd(m, ps−i+1 ) = 1 then [mui ]j = [ui ]π(j) where π ∈ Sps−i+1 . • If gcd(m, ps−i+1 ) = pl then [mui ]j = [ m u ] pl i+l π(j Sps−(i+l)+1 . mod ps−(i+l)+1 ) where π ∈ Proof. It is a straightforward comprobation. Theorem 2.3. Let n = pst1 +(s−1)t2 +...+ts −s and L(H) be the n × n matrix whose rows are the n possible linear combinations (with coefficients in Zps ) of the rows of At1 ,t2 ,...,ts . Then, H ∈ BH(n, ps ). Proof. By construction, the difference between two distinct rows of L(H) is a linear combination (with coefficients in Zps ) of the rows of At1 ,t2 ,...,ts . Hence, it is a row of L(H). Therefore, proving HH ∗ = nIn reduces to proving that every row sum of H is 0. For the rows of H corresponding to multiples of the rows of At1 ,t2 ,...,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 At1 ,t2 ,...,ts is by a simple induction. We provide some examples of Butson matrices coming from Theorem 2.3. Example 2.4. Let p = 2 and s = 3. For t1 = 1, t2 = 1, t3 = 0 then L(H) is the matrix given in Example 1.1. For t1 = 1, t2 = 1, t3 = 1 then 8 H ∈ BH(8, 8) where       L(H) =       0 0 0 0 0 0 0 0 0 2 4 6 0 2 4 6 0 4 0 4 0 4 0 4 0 6 4 2 0 6 4 2 0 0 0 0 4 4 4 4 0 2 4 6 4 6 0 2 0 4 0 4 4 0 4 0 0 6 4 2 4 2 0 6             Remark 2.5. Let L(H) be the matrix of Example 2.4 for t1 = 1, t2 = 1, t3 = 1. Then L(H) = L(F2 ⊗ F4 ) where we have used that F2 ⊗ F4 ∈ BH(8, 8) by means of ζ2 = ζ84 and ζ4 = ζ82 . In general we have the following. Proposition 2.6. Let n = pst1 +(s−1)t2 +...+ts −s and L(H) be the n × n matrix of Theorem 2.3. Then, H is equivalent to (Fp )ts ⊗ (Fp2 )ts−1 ⊗ . . . ⊗ (Fps−1 )t2 ⊗ (Fps )t1 −1 where Fps−j denotes the Fourier matrix of order ps−j embedded in BH(ps−j , ps ) j using that ζps−j = ζpps , and (M)r denotes the r-fold Kronecker product of the matrix M. Proof. The proof is by induction. The case t1 , t2 , . . . , ts = 1, 0, . . . , 0 is trivial, so consider the case t1 = 2 and  0. It is clear  t2 = . . . = ts = 0 0 ··· 0 . For the next that L(H) = L(Fps ) since A2,0...,0 = 0 1 · · · ps − 1 step of the induction, we assume that ti+1 = . . . = ts = 0 and L(H) = ′ L((Fps−(i−1) )ti ⊗ (Fps−(i−1) )ti−1 ⊗ . . . ⊗ (Fps−1 )t2 ⊗ (Fps )t1 −1 ). Now, we have to distinguish two possibilities: • t′i < ti ; then let t′i ← t′i + 1 and ti+1 = . . . = ts = 0. All the possible ′ linear combinations of the rows of At1 ,...,ti−1 ,ti +1,0...,0 are the rows of B = L(Fps−(i−1) ⊗ H). • t′i = ti ; then take ti+1 = 1 with ti+2 = . . . = ts = 0. 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. 9 Remark 2.7. For t1 6= 0, t2 = . . . = ts = 0 and p = 2, the code generated with the rows of At1 ,0,...,0 is a Zps -simplex code of type α (see [19, Definition 4.1]). Furthermore, this code is self-orthogonal if s = 2. Corollary 2.8. A simplex code of type α over Z2s of length 2sm (see [19]) and the code whose codewords are the rows of L((F2s )m ) are the same. Therefore Mψ , the cocyclic BH(2sm , 2s ) of [19, Theorem 5.1, ii)] is equivalent to (F2s )m . Similarly, when p > 2 prime, the analogous classifying result for the cocyclic BH(psm , ps ) of [21, Proposition 3.1, ii)] holds. Proof. Attending to the Remark above, a simplex code of type α over Z2s of length n = 2st1 is exactly the code FH 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 Znps is a Zps -additive code if it is a subgroup of n Zps (i.e., a Zps -module). Clearly, given a Zps -additive code, C, of length n there exist some non negative integers t1 , . . . , ts such that C is isomorphic (as an abelian group) to Ztp1s × Ztp2s−1 × . . . × Ztps . Thus, C is said to be of type (n; t1 , . . . , ts ). Note that |C| = pst1 p(s−1)t2 . . . pts since there are t1 (generators) codewords of order ps , t2 of order ps−1 and so on. Remark 2.9. Let t1 , . . . , ts be non negative integers and taking A1,0,...,0 = [1] instead of [0], the method described at the beginning of this section provides At1 ,t2 ,...,ts as a generator matrix for a Zps -additive code of type (n; t1 , . . . , ts ) where n = pst1 +(s−1)t2 +...+ts −s . The description of recursive constructions of these matrices are in [9, 13, 14] for p = 2. The case p 6= 2 has been studied in [24]. We will denote the codes associated to these matrices by Ht1 ,...,ts . Let us point out that H0,t2 ,...,ts ⊂ H1,t2 ,...,ts . Now, we establish the following result. Proposition 2.10. For t1 > 0, every Ht1 ,...,ts is a BH-code where the Butson Hadamard matrix is a Kronecker product of Fourier matrices. Proof. Let CH be the BH-code associated to H of Theorem 2.3. It is clear that CH is equivalent to Ht1 ,...,ts . Now, the result follows from Proposition 2.6. The following is an example of a BH-code which is not additive. 10 Example 2.11. Let H ∈ BH(8, 4)  0 0  0 1   0 3   0 0 L(H) =   0 2   0 3   0 1 0 2 with 0 3 2 1 0 3 2 1 0 0 1 1 2 2 3 3 0 2 0 2 0 2 0 2 0 3 3 2 2 1 1 0 0 1 2 3 0 1 2 3 0 2 1 3 2 0 3 1       .      CH is not Z22 -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 Zps -additive codes. Since any Zps -additive code is always of type (n; t1 , . . . , ts ) for some non negatives integers t1 , . . . , ts and Ht1 ,...,ts ∈ BH(pst1 +(s−1)t2 +...+ts −s , ps ), assuming that t1 > 0. 2.2 Generalized Gray map The Gray map is a function from Z4 to Z22 which is typically used to form binary codes from Z4 -codes. In what follows, we introduce a generalized Gray s−1 map Φp from Zps to Zpp , and extend this to a yet more general function ps−1 Ψp from Zmps to Zmp . For k = pe11 · · · pet t , and ℓ = p1 · · · pt the composition k/ℓ Ψpt · · · Ψp1 is a function from Zk to Zℓ . From this function we construct a morphism BH(n, k) → BH(nk/ℓ, ℓ). Where x = [x1 , . . . , xn ] ∈ Znk and ϕ is any function with domain Zk , we will write ϕ(x) = [ϕ(x1 ), . . . , ϕ(xn )]. Further, we write ϕ(C) = {ϕ(c) : c ∈ C} where C ⊆ Znk . We consider the elements of Zps−1 to be ordered in increasing lexicographic order. We denote by D the BH(ps−1 , p) matrix defined in Example 1.2 and label the rows of L(D) in the order 0, 1, . . . , ps−1 − 1. Let [L(D)]i denotes the s−1 row of L(D) labeled by i. Then we let Φp : Zps → Zpp be the map defined by Φp (x) = [L(D)]b + a1, x = aps−1 + b. Let us observe that for p = 2, Φp is the well-known Carlet’s map [4] and for p > 2, Φp is of type ϕ given in [24]. For what remains of this section we write Φ = Φp for brevity unless there is some confusion. Proposition 2.12 ([24]). The entrywise application of Φ is an isometric s−1 embedding of (Znps , d∗ ) into (Zpp n , dH ). Furthermore, if C is a code with parameters (n, M, d∗ ) over Zps , then the image code C = Φ(C) is a code with parameters (ps−1 n, M, dH ) over Zp . 11 Lemma 2.13. Let x, y ∈ Zps . Then Φ(x − y) = Φ(x) − Φ(y) + α1 where α ∈ {0, p − 1}. Proof. Let x = a1 ps−1 + b1 and y = a2 ps−1 + b2 . Then ( (a1 − a2 )ps−1 + (b1 − b2 ), if b1 ≥ b2 x−y = (a1 − a2 − 1)ps−1 + (b1 − b2 ), if b1 < b2 . Further, by the linearity of the inner product vw T and the definition L(D) = [vw T ]v,w∈Znp it follows that Φ(b1 − b2 mod ps−1 ) = [L(D)]b1 −b2 = [L(D)]b1 − [L(D)]b2 = Φ(b1 ) − Φ(b2 ). Thus Φ(x − y) = Φ(x) − Φ(y) + α1 where α = 0 if b1 ≥ b2 , and α = p − 1 otherwise. Given H ∈ Mn (hζps i), we write L(H Φ ) for Φ to  L(H)  L(H) + J   L(H) + 2J   ..  . L(H) + (ps−1 − 1)J the entrywise application of     .   Then H Φ is the corresponding matrix in Mnps−1 (hζp i). Theorem 2.14. If H ∈ BH(n, ps ), then H Φ ∈ BH(nps−1 , p). Proof. Observe that H Φ is Butson Hadamard over hζp i if, for all i 6= j, the sequence of differences [L(H Φ )]i,l − [L(H Φ )]j,l , 0 ≤ l ≤ n · ps−1 − 1 contains each element of Zp equally often. First note that for all i 6= j, the sequence of differences [L(H)]i,l − [L(H)]j,l , 0 ≤ l ≤ n − 1 contains each element of the s−1 form aps−1 equally often for a = 0, . . . , p − 1. This is a consequence of ζkap being a pth root of unity. By Lemma 2.13, if x − y = aps−1 then Φ(x − y) = Φ(x) − Φ(y). Since Φ(aps−1 ) = a1 for a ∈ Zp , 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 aps−1 , then the set of corresponding differences in [L(H Φ )]i,l −[L(H Φ )]j,l contains mps−1 repetitions of each element of Zp . Finally, if x − y 6≡ 0 mod ps−1, then Φ(x)−Φ(y) = Φ(x−y)+α1 for some α, where x−y = aps−1 +b and b 6= 0. Thus Φ(x − y) = a1 + [L(D)]b which contains every element of Zp exactly ps−2 -times, and so too does Φ(x) − Φ(y). Corollary 2.15. The image of any BH-code over Zps of length n by Φ is a BH-code over Zp of length n · ps−1 and minimum Hamming distance dH = nps−2 (p − 1). 12 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 Ht1 ,...,ts and p = 2). Proposition 2.17. Any BH-code CH of length n over Zps has minimum distance d∗ = nps−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 Fp (see [7, Lemma 2.2]). Thus, CH Φ = Φ(CH ) is a generalized Hadamard code as well since H Φ ∈ BH(ps−1 n, p). The minimum Hamming distance of these codes is well known to be nps−2 (p − 1). The fact that Φ is an isometric embedding (Proposition 2.12) concludes the proof. Now let k = mps where p does not divide m and recall that every element x ∈ Zk can be written uniquely as x = aps + bm mod k for some 0 ≤ a ≤ m − 1 and 0 ≤ b ≤ ps − 1. Then let Ψp (aps + bm) = mΦp (b) + ap1 s−1 define a map Zk → Zpmp . Proposition 2.18. The entrywise application of Ψp is an isometric embeds−1 ding of (Znmps , d† ) into (Zpmp n , dH ). Furthermore, if C is a code with parameters (n, M, d† ) over Zmps , then the image code C = Ψp (C) is a code with parameters (ps−1 n, M, dH ) over Zmp . Proof. This follows from a straight forward extension of Proposition 2.12. Given H ∈ Mn (hζk i) where k = ps m, we write L(H Ψp ) for the entrywise application of Ψp to   L(H)   L(H) + mJ     L(H) + 2mJ .    ..   . L(H) + (ps−1 − 1)mJ Then H Ψp is the corresponding matrix in Mnps−1 (hζpmi). We will devote the rest of this section to a proof of the following. Theorem 2.19. If H ∈ BH(n, k) where k = ps m, then H Ψp ∈ BH(nps−1 , pm). Repeated application of Ψp for all primes p dividing k gives the following. 13 Corollary 2.20. If there exists a BH(n, k) where k = ps11 · · · psrr , then there exists a BH(nk/ℓ, ℓ) where ℓ = p1 · · · pr . Before we can prove Theorem 2.19, we will need to establish some preliminary results. Hereafter we fix a prime p and let Ψ = Ψp . Lemma 2.21. For all 0 ≤ x, y < k = mps , Ψ(x − y) = Ψ(x) − Ψ(y) + mα1 where α ∈ {0, p − 1}. Proof. Let x = aps + bm and y = cps + dm. Observe that Ψ(x − y) = (a − c)p1 + mΦ(b − d). By Lemma 2.13, Φ(b − d) = Φ(b) − Φ(d) + α1 where α ∈ {0, p − 1}. The result follows. Pps−1 Ψ(z)i ω = Lemma 2.22. Let z 6= f ps−1 for any 0 ≤ f ≤ mp − 1. Then i=1 0 where ω is a primitive k th root of unity. Otherwise, Ψ(z) = f 1, and Pps−1 Ψ(z)i = ps−1 ω f . i=1 ω Proof. First suppose that z 6= f ps−1. Observe that Ψ(z) = m[L(D)]j + α1 Pps−1 [L(D)]j,i +α Pps−1 −1 Ψ(z)i ω = ω = i=1 for some α ∈ Zpm and j 6= 0. Then i=0 Pps−1 −1 [L(D)]j,i α ω ω = 0. i=0 Now suppose that z = f ps−1 . Then f = gm + hp mod mp where 0 ≤ g ≤ p − 1 and 0 ≤ h ≤ m − 1. Thus f ps−1 = hps + gmps−1 mod ps m. It follows that Ψ(z) = hp1 + mΦ(gps−1 ) = hp1 + gm1 = f 1. Corollary 2.23. If x = f ps−1 and y 6= 0 mod ps−1 , then Ψ(x − y) = Ψ(x) − Ψ(y) + m(p − 1)1. Consequently, for any multiset X of elements of Zk such that x ∈ X only if x = f ps−1,and for any y 6= 0 mod ps−1 , then P Pps−1 Ψ(x−y)i = 0. i=1 ω x Proof. Since x = f ps−1 , by Lemma 2.22 we have Ψ(x) = f 1. Since y = Pps−1 Ψ(y)i ω = 0. Complex cps + dm 6= 0 mod ps−1 , by Lemma 2.22 we have i=1 Pps−1 −Ψ(y)i = 0. conjugation is a field automorphism so it follows too that i=1 ω It follows from Lemma 2.21 that Ψ(x − y) = Ψ(x) − Ψ(y) + m(p − 1)1, and P s−1 P s−1 so pi=1 ω Ψ(x−y)i = ω f +m(p−1) pi=1 ω −Ψ(y)i = 0. We will require the following result of Lam and Leung. Lemma 2.24 (Corollary 3.2, [15]). If α1 +· · ·+αr = 0 is a minimal vanishing sum of nth roots of unity, then after a suitable rotation, we may assume that all αi ’s are nth 0 roots of unity where n0 is square-free. 14 The sum α1 + · · · + αr = 0 is minimal if no proper subsums can be zero. A rotation in this context is a multiplication of the sum by an nth root of unity. P Suppose that for some multiset X of elements of Zk , we have that x ω x = 0 is minimal, and further assume that each ω x is an nth 0 root of unity for n0 s−1 square-free. Then for each x ∈ X, x = f p for some f . Lemma 2.22 P Pps−1 Ψ(x)i = 0, and then applying Corollary 2.23, we get implies that x i=1 ω P Pps−1 Ψ(x−y)i = 0 for all y 6= 0 mod ps−1 . Any vanishing sum with that x i=1 ω th terms that are not n0 roots of unity can only be scaled so that the terms y s−1 are all nth . Thus we prove 0 roots of unity by some ω where y 6= 0 mod p the following. P P Pps−1 Ψ(x)i Lemma 2.25. If x ω x = 0 is minimal, then x i=1 ω = 0. Proof. If the terms ω x are nth 0 roots of unity then this is immediate from Lemma 2.22. Otherwise, we scale by some ω y such that y 6= 0 mod ps−1 so that the terms are then nth 0 roots of unity. Then again we apply Lemma 2.22 and prove the original equality using Corollary 2.23. Finally, we can prove Theorem 2.19. Proof. Observe that the rows of H Ψ can be partitioned into ps−1 blocks of size n corresponding to the images of the rows of L(H)+rmJ for 0 ≤ r ≤ ps−1 −1. Given H ∈ BH(n, k), the Hermitian inner product of two distinct rows is zero. That is, for any two distinct rows x = [x1 , . . . , xn ] and y = [y1 , . . . , yn ] of L(H), the Hermitian inner product of the corresponding rows of H is of the form n X ω xi−yi = 0. i=1 Since we can partition this equation into minimal sums, it follows that Pn Pps−1 (Ψ(xi )−Ψ(yi ))j = 0. That is, distinct rows of H Ψ from each j=1 ω i=1 block of n rows are pairwise orthogonal. To see that two rows taken from distinct blocks are orthogonal, we observe that tm 6= 0 mod ps−1 for any 1 ≤ t ≤ ps−1 − 1, and so we also apply Corollary 2.23. Remark 2.26. The application of the map Ψ2 to H ∈ BH(n, 4) is equivalent to a familiar morphism BH(n, 4) → BH(2n, 2) of Turyn [26]. That is, for any H ∈ BH(n, 4), the Hadamard matrix obtained from Turyn’s morphism applied to H is Hadamard equivalent to H Ψ2 . By Proposition 2.18 we know that d† (x, y) = dH (Ψ(x), Ψ(y)). We may also relate the minimum Hamming distance of Ψ(C) directly to the minimum Hamming distance of C, but less precisely. 15 Proposition 2.27. Let C be a BH-code of minimum Hamming distance d obtained from a BH(n, ps m) with p a prime not dividing m. Then the minimum distance d′ of Ψ(C) is in the range d(p − 1)ps−2 ≤ d′ ≤ dps−1 . Proof. If xi 6= yi , then ps−1 − ps−2 ≤ dH (Ψ(xi ), Ψ(yi )) ≤ ps−1 . Hence dH (x, y)(p − 1)ps−2 ≤ dH (Ψ(x), Ψ(y)) ≤ dH (x, y)ps−1 . 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(32). 3 Propelinear codes and cocyclic matrices The BH-matrix given in Example 2.11, H, is cocyclic over Z8 and its BHcode associated CH is not linear. Can we define a propelinear structure in CH ? 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 : G × G → U such that ψ(g, h)ψ(gh, k) = ψ(g, hk)ψ(h, k) ∀ g, h, k ∈ G (3) 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)−1φ(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 Eψ 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 Eψ and the set T (ψ) = {(1, g) : g ∈ G} is a normalized transversal of U × {1} in Eψ . In the other direction, suppose that E is a finite group with normalized transversal T for a central subgroup U. Put G = E/U and σ(tU) = t for t ∈ T . The map ψT : G × G → U defined by ψT (g, h) = σ(g)σ(h)σ(gh)−1 is a cocycle; furthermore, EψT ∼ = E. Each cocycle ψ ∈ Z 2 (G, U) is displayed as a cocyclic matrix Mψ : under some indexing of the rows and columns by G, Mψ has entry ψ(g, h) in position (g, h). A n × n matrix A = (ag,h )g,h∈G is called G-invariant (or just group invariant) if agk,hk = ag,h for all g, h, k ∈ G. 16 −1 Lemma 3.1. If A is G-invariant and ag,h ∈ U then ψ(g, h) = a−1 g,0 ag,h−1 a0,h−1 is a cocycle. Remark 3.2. Every group invariant matrix with entries in U is equivalent to a cocyclic matrix. 2 Fixing U = hζ k i. A cocycle ψ ∈ Z (G, hζk i) is called orthogonal if, for P each g 6= 1 ∈ G, h∈G ψ(g, h) = 0. Proposition 3.3. [12] Hψ ∈ BH(n, k) if and only if ψ ∈ Z 2 (G, hζk i) is orthogonal. Fact: A cocyclic Butson Hadamard matrix is not necessarily pairwise row and column balanced. Proposition 3.4. Given ψ ∈ Z 2 (G, hζk i) and x = ζkλ [ψ(g, g1), . . . , ψ(g, gn)] for a fixed order in G = {g1 = 1, g2 , . . . , gn }. Define πx ∈ Sn so that πx−1 (j) = k where gk = ggj . Then 1. x + πx (y) = ζkλ+µ ψ(h, g) [ψ(hg, g1), . . . , ψ(hg, gn )] where + means the componentwise product and y = ζkµ [ψ(h, g1 ), . . . , ψ(h, gn )]. 2. πx+πx (y) = πx (πy ). Proof. 1. Observe that πx (y) = ζkµ [ψ(h, gg1), . . . , ψ(h, ggn)]. Hence the ith component of x + πx (y) is ζkλ+µ ψ(g, gi )ψ(h, ggi ). Apply (3) letting (g, h, k) = (h, g, gi ) and the result follows. 2. Let z = ζkγ [ψ(ℓ, g1 ), . . . , ψ(ℓ, gn )]. From part 1 we know that x + πx (y) is a scalar multiple of the n-tuple defined by ψ(hg, −), and thus the j th component of πx+πx (y) (z) is ψ(ℓ, hggj ). Now observe that the k th component of πy (z) is ψ(ℓ, hgk ). We have πx (k) = j where gk = ggj , and thus the j th component of πx (πy (z)) is ψ(ℓ, hgk ) = ψ(ℓ, hggj ). Corollary 3.5. Let ψ ∈ Z 2 (G, hζk i) and Hψ ∈ BH(n, k). Then the corresponding BH-code CH is a BHFP-code where x ⋆ y = x + πx (y) for all x, y ∈ C. Proof. Extend the definition of πx for the rows x of L(Hψ ) to all of CH by letting πx+α1 = πx for all α ∈ Zk . The code CH is propelinear by Proposition 3.4, and since x ⋆ y = x + πx (y) for all x, y ∈ C, the first property of Definition 1.5 is satisfied. Finally observe that because πx ∈ Sn is defined so that πx−1 (j) = k where gk = ggj , it follows that πx fixes no coordinate when x 6= α1, and πα1 = IdSn for all α ∈ Zk . 17 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 Cp when p is prime. Corollary 3.7. Let CH be a BHFP-code of length n over Zk 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. CH = ∪1≤α≤n Cα where Cα = {[L(H)]α +λ1}λ∈Zk 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 Z8 . Then, CH = C1 ∪ C2 ∪ . . . ∪ C8 can be endowed with a full propelinear structure with the following group Π of permutations  I x ∈ C1     (1, 2, 3, 4, 5, 6, 7, 8) x ∈ C2     (1, 3, 5, 7)(2, 4, 6, 8) x ∈ C3    (1, 4, 7, 2, 5, 8, 3, 6) x ∈ C4 πx = (1, 5)(2, 6)(3, 7)(4, 8) x ∈ C5     (1, 6, 3, 8, 5, 2, 7, 4) x ∈ C6     (1, 7, 5, 3)(2, 8, 6, 4) x ∈ C7    (1, 8, 7, 6, 5, 4, 3, 2) x ∈ C8 18 CH is a BHFP-code with group structure Z8 ×Z4 and Π ∼ = Z8 . The codewords are C1 C2 C3 C4 C5 C6 C7 C8 = {[0, 0, 0, 0, 0, 0, 0, 0] + λ1}, = {[0, 1, 3, 0, 2, 3, 1, 2] + λ1}, = {[0, 3, 2, 1, 0, 3, 2, 1] + λ1}, = {[0, 0, 1, 1, 2, 2, 3, 3] + λ1}, = {[0, 2, 0, 2, 0, 2, 0, 2] + λ1}, = {[0, 3, 3, 2, 2, 1, 1, 0] + λ1}, = {[0, 1, 2, 3, 0, 1, 2, 3] + λ1}, = {[0, 2, 1, 3, 2, 0, 3, 1] + λ1} where λ runs through Z4 , and CH is a (8, 32, 4)-code over Z4 . CH has a group structure Z8 × Z4 ≃ ha, 1 | a8 = 14 = 0i, where a = [0, 1, 3, 0, 2, 3, 1, 2]. An interesting family of BH-codes over Zps are those associated to Kronecker products of Fourier matrices. They are denoted by Ht1 ,t2 ,...,ts (see Remark 2.9 and Proposition 2.10) and since these matrices are cocyclic over t G = Ztps × Zps−1 × . . . × Ztp2s−1 × Ztp1s−1 , these codes can be endowed with 2 a full propelinear structure by Corollary 3.5 . Furthermore, for p = 2 and s = 2 in [20], it is shown that the image of Ht1 ,t2 under the Gray map are in fact propelinear codes. Example 3.9. Considering H1,1,1 , the Z8 -additive code of length n = 8 associated to L(H) of Example 2.4. Then, it can be endowed with a full propelinear structure with the following group Π of permutations Π ∼ = Z2 × Z4 generated by πx and πy where x = [0, 2, 4, 6, 0, 2, 4, 6], y = [0, 0, 0, 0, 4, 4, 4, 4], πx = (1, 4, 3, 2)(5, 8, 7, 6), πy = (1, 5)(2, 6)(3, 7)(4, 8). The full propelinear code is a group (H1,1,1 , ⋆) ∼ = Z8 ×Z4 ×Z2 = hx, y, 1 | x8 = 0, y2 = 14 = x4 i. 4 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, mps ), is the same as the number of codewords in the BH-code C ′ 19 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 Z9 -code C obtained from the trivial BH(1, 9), and the Z3 code Ψ(C) obtained from the BH(3, 3) matrix H ′ = (1)Ψ which written in log form is   0 0 0 L(H ′ ) =  0 1 2  0 2 1 The code C is clearly linear, and as a group is isomorphic to the cyclic group Z9 . 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 ∼ = Z9 generated by any single element x, no matter what the coordinate permutation πx 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 πx = (1, 3, 2). It is isomorphic to Z23 . However, we find that for the special case Ψ2 : Z4m → Z22m , we can carefully construct an isomorphism between the groups of codewords C and C ′ = Ψ2 (C), and determine the group operation ⋆′ so that (C, ⋆) ∼ = (C ′ , ⋆′ ). Let Ψ = Ψ2 . Theorem 4.1. Let m be an odd positive integer, and let C ⊆ Zn4m be a full propelinear code. Then the code C ′ = Ψ(C) is full propelinear with group structure (C ′ , ⋆′ ) ∼ = (C, ⋆). 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 ρx ∈ S2 for each x ∈ Z4m so that Ψ(x) + ρx (Ψ(y)) = Ψ(x + y) for all y. We will see that ρx = (1, 2)x , i.e., ρx permutes the two coordinates of a word in Z22m 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) + ρx (Ψ(y)) = Ψ(x + y) for each pair (b, d) ∈ Z4 , but for brevity we take (b, d) = (3, 1) as an example 20 and leave the rest to the reader. Observe that Ψ(x) = [2a, 2a] + mΦ(3) = [2a, 2a] + m([0, 1] + [1, 1]) = [2a + m, 2a], Ψ(y) = [2c, 2c] + mΦ(1) = [2c, 2c] + m([0, 1] + [0, 0]) = [2c, 2c + m], Ψ(x + y) = [2(a + c), 2(a + c)] + mΦ(0) = [2(a + c), 2(a + c)]. Since b = 3, x is odd, and so ρx = (1, 2). It follows that Ψ(x) + ρx (Ψ(y)) = Ψ(x + y). This verifies the 1-dimensional case. Now suppose that C is full propelinear of length n, and let x, y ∈ C, with x ⋆ y = x + πx (y). Let πΦ(x) ∈ S2n permute the n blocks of size 2, labelled b1 , . . . , bn , according to the action of πx on a word of length n. That is, πΦ(x) (bi ) = bj if and only if πx (i) = j. Then πΦ(x) (Ψ(y)) = Ψ(πx (y)). Further, let ρi = (2i − 1, 2i) be swapping the entries of Qnthe xpermutation i the block bi , and write ρx = i=1 ρi . It follows that Ψ(x) ⋆′ Ψ(y) := Ψ(x) + ρx πΦ(x) (Ψ(y)) = Ψ(x + πx (y)) = Ψ(x ⋆ y). Thus Ψ is a bijective homomorphism from (C, ⋆) to (C ′ , ⋆′ ). It remains to verify that the permutation ρx πΦ(x) = IdS2n whenever Ψ(x) = α12n for any α ∈ Z2m , and has no fixed coordinate otherwise. Let S = C ∩ {α1n : 0 ≤ α ≤ 4m − 1} and let X ⊂ S be the subset X = C ∩ {2α1n : 0 ≤ α ≤ 2m − 1}. Note first that Ψ(X) is the set X ′ = C ′ ∩ {α12n : 0 ≤ α ≤ 2m − 1}. It is clear that ρx πΦ(x) = IdS2n for all x ∈ X. Further, for any s ∈ S \ X, ρs = (1, 2)(3, 4) · · · (2n − 1, 2n), and so does not fix any coordinate. Finally, for any codeword c ∈ C \ S, πc does not fix any coordinate of Zn4m , and it follows that πΦ(c) does not fix any coordinate of Z2n 2m . Corollary 4.2. Let m be an odd positive integer, and let H ∈ BH(n, 4m). If the BH-code C obtained from H is full propelinear with group structure G, then the BH-code C ′ obtained from H Ψ where Ψ is full propelinear with group structure G′ ∼ = G. Example 4.3. Let H3,0 be the BH-code associated to F4 ⊗ F4 ∈ BH(16, 4) and H 3,0 be its image by the Gray map which is known to be a nonlinear code (see [9, Table 1]). H3,0 is full propelinear, with permutation group Π ∼ = Z24 generated by πx and πy where x = [0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3], y = [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3], πx = (1, 4, 3, 2)(5, 8, 7, 6)(9, 12, 11, 10)(13, 16, 15, 14), πy = (1, 13, 9, 5)(2, 14, 10, 6)(3, 15, 11, 7)(4, 16, 12, 8). 21 The corresponding permutations ρx πΦ(x) and ρy πΦ(y) are as follows: ρx πΦ(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), ρy πΦ(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). Thus, H 3,0 can be endowed with a full propelinear structure with the group hρx πΦ(x) , ρy πΦ(y) i of permutations, which is non-abelian of order 32. This group contains the element (ρx πΦ(x) )(ρy πΦ(y) )(ρx πΦ(x) )−1 (ρy πΦ(y) )−1 = ρ1 πΦ(1) = (1, 2)(3, 4) · · · (31, 32). The groups (H3,0 , ⋆) ∼ = (H 3,0 , ⋆′) are isomorphic to Z2 × Z4 × Z8 . Remark 4.4. Even though the codes C and C ′ are isomorphic as groups according to Theorem 4.1, the example above shows that the underlying groups of coordinate permutations are not necessarily isomorphic. As a simpler example, take the trivial 1-dimensional Z4 code and its image in Z22 . Here, Ψ : [0], [1], [2], [3] 7→ [0, 0], [0, 1], [1, 1], [1, 0]. Both are cyclic, generated by [1] and [0, 1] respectively, but the group of coordinate permutations of Z4 is necessarily trivial, and the group of coordinate permutations of the image is generated by ρ[1] π[0,1] = (1, 2). More generally, if C is a BHFP-code obtained from a BH(n, 4m) with group Π of coordinate permutations then by Definition 1.5, |Π| = n, and the group of coordinate permutations for Ψ(C) will be of order |Π′ | = 2n. Acknowledgements The authors would also like to thank Kristeen Cheng for her reading of this manuscript. The first author was supported by the project FQM-016 funded by JJAA (Spain). The second author was supported by the Spanish grant TIN2016-77918-P (AEI/FEDER, UE). The third author was supported by the Irish Research Council (Government of Ireland Postdoctoral Fellowship, GOIPD/2018/304). References [1] Armario, J.A., Bailera, I., Egan R.: Generalized Hadamard full propelinear codes. ArXiv:1906.06220 [math.CO], (2019). [2] Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, (2006). 22 [3] Borges, J., Mogilnykh, I.Y., Rifà, J. Solov’eva, F.: On the number of nonequivalent propelinear extended perfect codes. Electronic J. Combinatorics 20, 1–14 (2013). [4] Carlet, C.: Z2k -linear codes. IEEE Trans. Inf. Theory 44, 1543–1547 (1998). [5] Constantinescu, I., Heise, W.: A metric for codes over residue class rings of integers. Problems Inf. Transmiss. 33, 22–28 (1997). [6] Duc, T. D., Schmidt, B.: Bilinear Forms on Finite Abelian Groups and Group Invariant Butson Hadamard Matrices. J. Comb. Theory Ser. A 166, 337-351 (2019). [7] Egan, R., Flannery, D.L., Ó Catháin, P.: Classifying cocyclic Butson Hadamard matrices. In: Colbourn, C. (Ed.) Algebraic Design Theory and Hadamard Matrices, Springer Proc. Math. Stat. 133, 93–106 (2015). [8] Egan, R., Ó Catháin, P.: Morphisms of Butson classes. Linear Algebra Appl. 577, 78–93 (2019). [9] Fernández-Córdoba, C., Vela, C., Villanueva, M.: On Z2s -linear Hadamard codes: kernel and partial classification. Des. Codes and Cryptogr. 87, 417–435 (2019). [10] Greferath, M., McGuire, G., O’Sullivan, M.: On Plotkin-optimal codes over finite Frobenius rings. J. Algebra Appl. 5, 799–815. [11] Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The Z4 -linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40 (2), 301–319 (1994). [12] Horadam, K.J.: Hadamard Matrices and Their Applications. Princeton University Press, Princeton, NJ (2007). [13] Krotov, D. S.: Z4 -linear Hadamard and extended perfect codes. International workshop on coding and cryptography, ser. Electron. Notes Discret. Math. 6, 107–112 (2001). [14] Krotov, D. S.: On Z2k -dual binary codes. IEEE Trans. Inf. Theory 53, 1532–1537 (2007). [15] Lam, T. Y., Leung, K. H.: On vanishing sums of roots of unity. J. Algebra 224(1), 91–109 (2000). 23 [16] Lampio, P., Östergård, P., Szöllósi, F.: Orderly generation of Butson Hadamard matrices. Math. Comp. 89, 313–331 (2020). [17] McGuire, G., Ward, H.: Cocyclic Hadamard matrices from forms over finite Frobenius rings. Linear Algebra Appl. 430, 1730–1738 (2009). [18] Ó Catháin, P., Swartz, E.: Homomorphisms of matrix algebras and constructions of Butson-Hadamard matrices. Disc. Math. 342(12), 111606 (2019). [19] Pinnawala, N., Rao, A.: Cocyclic simplex codes of type α over Z4 and Z2s . IEEE Trans. Inf. Theory 50, 2165–2169 (2004). [20] Pujol, J., Rifà, J.: Translation invariant propelinear codes. IEEE trans. Inf. Theory 43, 590–598 (1997). [21] Rao, A., Pinnawala, N.: New linear codes over Zps via the trace map. 2005 IEEE International Symposium on Information Theory (Adelaide, Australia), 4–9 September 2005, pp. 124–126. [22] Rifà, J., Basart, J.M., Huguet, L.: On completely regular propelinear codes. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. LNCS 357, pp. 341–355. Springer, Berlin (1989). [23] Rifà, J., Suárez, E.: Hadamard full propelinear codes of type Q. Rank and kernel. Des. Codes Cryptogr. 86, 1905–1921 (2018). [24] Shi, M., Wu, R., Krotov, D. S.: On Zp Zpk -additive codes and their duality. IEEE trans. Inf. Theory 65, 3841–3847 (2019). [25] Schmidt, B.: A Survey of Group Invariant Butson Matrices and Their Relation to Generalized Bent Functions and Various Other Objects. Radon Series on Computational and Applied Mathematics 23, 241-251 (2019). [26] Turyn, R. J.: Complex Hadamard matrices. In: Combinatorial Structures and Their Applications, Proc. Calgary Internat. Conf., Calgary, Alta., 1969, Gordon and Breach, New York, 435–437, (1970). 24