Butson full propelinear codes

Butson full propelinear codes José Andrés Armario∗ Universidad de Sevilla, Sevilla, Spain Ivan Bailera† Universitat Autònoma de Barcelona, Bellaterra, Spain Ronan Egan‡ National University of Ireland Galway, Galway, Ireland November 30, 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 ℓdividing k, provided that any prime pdividing kalso divides ℓ. We prove that a Zps-additive code with pa prime number is isomorphic as a group to a BH-code over Zpsand 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:[email protected] †E-mail:[email protected] ‡E-mail:[email protected] 1 1 Introduction Let nand kbe positive integers, and ζk= exp (2π√−1/k)be the complex kth root of unity. We write hζki={ζj k}0≤j≤k−1. Let Zkbe the ring of integers modulo kwith k > 1, and denote by Zn kthe set of n-tuples over Zk. We use bold notation x= [x1,...,xn]∈Zn kto denote vectors (or codewords) in Zn k. We denote the set of n×nmatrices with entries in a set Xby Mn(X). 1.1 Butson Hadamard matrices AButson Hadamard (or simply Butson) matrix of order nand phase kis a matrix H∈ Mn(hζki)such that HH∗=nIn, where Indenotes the identity matrix of order nand 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 Fn= [ζ(i−1)(j−1) n]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×nmonomial matrix with non-zero entries in the kth roots of unity. For any pair of such monomial matrices (P, Q)the operation defined by H(P, Q) = PHQ∗=H′is an equivalence operation, and Hand 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= [ζϕi,j k]n i,j=1 is represented by the matrix L(H) = [ϕi,j mod k]n i,j=1 with the convention that Li,j ∈Zkfor all i, j ∈ {1,...,n}. Example 1.1. he following is a BH(4,8) matrix H, display in logarithmic form L(H) =     0000 0246 0404 0642     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 pbe a prime number. If L(D) = [xyT]x,y∈Zn pthen Dis a BH(pn, p). In fact Dis 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 > 2only 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=pe1 1···pet tand m=pe1−1 1···pet−1 t, 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/√nis 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 Z4under 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 Zkcode) of length nis a nonempty subset Cof Zn k. The elements of Care 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∈Zn k, 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=ydH(x,y)for a code Cof length n, we say Cis a(n, |C|, d)code. Other distances functions are used, for instance, the Lee distance between two vectors x,y∈Zn kis dL(x,y) = wtL(x−y)where the Lee weight of a vector z= [z1,...,zn]∈Zn kis wtL(z) = Pn i=1 wtL(zi)with wtL(zi) = min{zi, k −zi}. Given H∈BH(n, k), we denote by FHthe Zk-code of length nconsisting of the rows of L(H), and by CHthe Zk-code defined as CH=∪α∈Zk(FH+α1) where 1denotes the all-one vector (and α1the all-αvector). We will write 1nto denote the all-one vector of length nwhen clarification is required. The code CHover Zkis 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]}, CH=                        [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], [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 Zn kis given by a coordinate permutation πand npermutations σ1,...,σnof Zk. We denote by Aut(Zn k)the group of all isometries of Zn k: Aut(Zn k) = {(σ, π) : σ= (σ1,...,σn)with σi∈Sym Zk, π ∈ Sn} where Sym Zkand Sndenote, respectively, the symmetric group of permutations on Zkand on the set {1,...,n}. The action of (σ, π)is defined as (σ, π)(v) = σ(π(v)) for any v∈Zn k, and the group operation in Aut(Zk)is the composition (σ, π)◦(σ′, π′) = ((σ1◦σ′ π−1(1),...,σn◦σ′ π−1(n)), π ◦π′) for all (σ, π),(σ′, π′)∈Aut(Zk). Definition 1.4. A code Cof length nover Zkhas a propelinear structure if for any codeword x∈Cthere exist πx∈ Snand σx= (σx,1,...,σx,n)with σx,i ∈Sym Zksatisfying: (i) (σx, πx)(C) = Cand (σx, πx)(0) = x, (ii) if y∈Cand 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⊆Zn k, we denote by Aut(C)the group of all isometries of Zn kfixing the code Cand we call it the automorphism group of the code C. A code Cover Zkis 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 Chas 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∈Zn k. The vector 0is always a codeword where π0=Idnis the identity coordinate permutation and σ0,i =Idkis the identity permutation on Zkfor all i∈ {1,...,n}. Hence, 0is the identity element in Cand πx−1=π−1 xand σx−1,i =σ−1 x,πx(i)for all x∈Cand for all i∈ {1,...,n}. We call (C, ⋆)apropelinear code. Henceforth we use Cinstead of (C, ⋆)if there is no confusion. Definition 1.5. A full propelinear code is a propelinear code Csuch that for every a∈C,σa(x) = a+xand πahas not any fixed coordinate when a6=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 Fqwhere qis 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 xand y. We define other useful distance functions here. Initially, let k=psfor a prime p. The weight function wt∗(x)with x∈Zpsis defined by wt∗(x) =    (p−1)ps−2x6=kps−1mod ps, k ∈Zp ps−1x=kps−1mod ps, k ∈Zp\{0} 0x= 0 mod ps For p=s= 2, this is the Lee weight. The corresponding distance d∗on Zn psis defined as follows: d∗(x,y) = n X i=1 wt∗(yi−xi),(1) where x= [x1, . . ., xn]and y= [y1,...,yn]in Zn ps. More generally, let k=mpsfor mcoprime to p. Any x∈Zkmay be written uniquely in the form x=aps+bm mod kwhere 0≤a≤m−1and 0≤b≤ps−1. Define the weight function wt†(x)on Zkby wt†(x) = wt∗(b)a= 0 ps−1a6= 0. The definition of the weight function here is consistent with the homogeneous metric introduced in [5]. The corresponding distance d†on Zn mpsis defined as follows: d†(x,y) = n X i=1 wt†(yi−xi),(2) where x= [x1,...,xn]and y= [y1,...,yn]in Zn mps. Given H∈BH(n, k), recall that FHis the Zk-code of length nconsisting of the rows of L(H), and CH=∪α∈Zk(FH+α1). Let ri H(l)be the number of repetitions of l∈Zkin the i-th row of L(H)and rH(l) = max 2≤i≤nri H(l). For k=ps, Lemma 3.1 of [16] gives a pattern that any row of L(H) has to follow. That is, any row xhas to be a permutation of the vector (u, r11+u,...,rt−11+u)where u= [0, ps−1,2ps−1,...,,(p−1)ps−1],ri∈ Zps−1,for 1≤i≤t−1with t=n p. Therefore, rH(l)≤n pl=hps−1where h∈Zp n p−1Otherwise. 6 As a consequence, n−n pis an upper bound for the minimum Hamming distance of FHwhen k=ps. Furthermore, the minimum Hamming distance of both codes, FHand CH, is the same in this case. In [19, 21], the authors prove that if n=psm and k=psthen the minimum Hamming distance of FHis n−n pand the minimum Lee distance is given by dL=2m+s−2, p = 2 ps(m+1)−2 4(p2−1), p > 2prime; where His the Butson matrix of Theorem 2.3 and m=t1−1for t1>0and 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 FHhas parameters (n−1, n, γn)meeting the Plotkin bound over Frobenius rings where γis 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−sand phase k=ps, where pis a prime. Let s be a positive integer, t1, t2,...,tsbe nonnegative integers with t1≥1, and A1,0,...,0= [0]. The matrix At1,t2,...,ts, where pi−1denotes the all-pi−1vector, 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< tido A←At′ 1,...,t′ s t′ i←t′ i+ 1 At′ 1,...,t′ s←Ai=A A . . . A 0·pi−11·pi−1... (ps−i+1 −1) ·pi−1 end while end for By construction, it is clear that At1,t2,...,tsis 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. Example 2.1. For p= 2 and s= 3. We have A1,1,0=0 0 0 0 0 2 4 6 and 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∈Zn k, the order of xis the smallest positive integer msuch that mx=0over Zk. Lemma 2.2. Let k=psand ui= [ 0·pi−1,1·pi−1, . . . , (ps−i+1 −1) ·pi−1]∈ Zps−i+1 pswhere 1≤i≤s. Then • ps−i+1−1 X j=0 ζjpi−1 k= 0, for all 1≤i≤s. •The order of uiis 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) = plthen [mui]j= [m plui+l]π(jmod ps−(i+l)+1)where π∈ Sps−(i+l)+1 . Proof. It is a straightforward comprobation. Theorem 2.3. Let n=pst1+(s−1)t2+...+ts−sand L(H)be the n×nmatrix whose rows are the npossible 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∗=nInreduces to proving that every row sum of His 0. For the rows of Hcorresponding 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 Hcorresponding to a linear combination of the rows of At1,t2,...,tsis 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) =             00000000 02460246 04040404 06420642 00004444 02464602 04044040 06424206             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=ζ4 8and ζ4=ζ2 8. In general we have the following. Proposition 2.6. Let n=pst1+(s−1)t2+...+ts−sand L(H)be the n×nmatrix of Theorem 2.3. Then, His equivalent to (Fp)ts⊗(Fp2)ts−1⊗... ⊗(Fps−1)t2⊗(Fps)t1−1 where Fps−jdenotes the Fourier matrix of order ps−jembedded in BH(ps−j, ps) using that ζps−j=ζpj ps, and (M)rdenotes the r-fold Kronecker product of the matrix M. Proof. The proof is by induction. The case t1, t2,...,ts= 1,0,...,0is trivial, so consider the case t1= 2 and t2=... =ts= 0. It is clear that L(H) = L(Fps)since A2,0...,0=0 0 ··· 0 0 1 ··· ps−1. For the next step of the induction, we assume that ti+1 =... =ts= 0 and L(H) = L((Fps−(i−1) )t′ i⊗(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,t′ i+1,0...,0are 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 Proposition 2.27. Let Cbe a BH-code of minimum Hamming distance dobtained from a BH(n, psm) with pa 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 xi6=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 Cobtained 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 Z8and its BH- code associated CHis not linear. Can we define a propelinear structure in CH? Certainly, we can and this is not an isolated situation. Let Gand Ube finite groups, with Uabelian, of orders nand k, respectively. A map ψ:G×G→Usuch 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 ψis 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)−1φ(gh) is a coboundary. The set of all cocycles ψ:G×G→Uforms an abelian group Z2(G, U)under pointwise multiplication. Factoring out the subgroup of coboundaries gives H2(G, U), the second cohomology group of Gwith coefficients in U. Given a group Gand ψ∈Z2(G, U), denote by Eψthe canonical central extension of Uby 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 Ulies in the centre of Eψand the set T(ψ) = {(1, g) : g∈G}is a normalized transversal of U× {1}in Eψ. In the other direction, suppose that Eis a finite group with normalized transversal Tfor a central subgroup U. Put G=E/U and σ(tU) = tfor t∈T. The map ψT:G×G→Udefined by ψT(g, h) = σ(g)σ(h)σ(gh)−1is a cocycle; furthermore, EψT∼ =E. Each cocycle ψ∈Z2(G, U)is displayed as a cocyclic matrix Mψ: under some indexing of the rows and columns by G,Mψhas entry ψ(g, h)in position (g, h). An×nmatrix A= (ag,h)g,h∈Gis called G-invariant (or just group invariant) if agk,hk =ag,h for all g, h, k ∈G. 16 Lemma 3.1. If Ais G-invariant and ag,h ∈Uthen ψ(g, h) = a−1 g,0ag,h−1a−1 0,h−1 is a cocycle. Remark 3.2. Every group invariant matrix with entries in Uis equivalent to a cocyclic matrix. Fixing U=hζki. A cocycle ψ∈Z2(G, hζki)is called orthogonal if, for each g6= 1 ∈G, Ph∈Gψ(g, h) = 0. Proposition 3.3. [12] Hψ∈BH(n, k)if and only if ψ∈Z2(G, hζki)is orthogonal. Fact: A cocyclic Butson Hadamard matrix is not necessarily pairwise row and column balanced. Proposition 3.4. Given ψ∈Z2(G, hζki)and x=ζλ k[ψ(g, g1),...,ψ(g, gn)] for a fixed order in G={g1= 1, g2,...,gn}. Define πx∈ Snso that π−1 x(j) = kwhere gk=ggj. Then 1. x+πx(y) = ζλ+µ kψ(h, g) [ψ(hg, g1),...,ψ(hg, gn)] where +means the componentwise product and y=ζµ k[ψ(h, g1),...,ψ(h, gn)]. 2. πx+πx(y)=πx(πy). Proof. 1. Observe that πx(y) = ζµ k[ψ(h, gg1),...,ψ(h, ggn)]. Hence the ith component of x+πx(y)is ζλ+µ kψ(g, gi)ψ(h, ggi). Apply (3) letting (g, h, k) = (h, g, gi)and the result follows. 2. Let z=ζγ k[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 jth component of πx+πx(y)(z)is ψ(ℓ, hggj). Now observe that the kth component of πy(z)is ψ(ℓ, hgk). We have πx(k) = jwhere gk=ggj, and thus the jth component of πx(πy(z)) is ψ(ℓ, hgk) = ψ(ℓ, hggj). Corollary 3.5. Let ψ∈Z2(G, hζki)and Hψ∈BH(n, k). Then the corresponding BH-code CHis a BHFP-code where x⋆y=x+πx(y)for all x,y∈ C. Proof. Extend the definition of πxfor the rows xof L(Hψ)to all of CHby letting πx+α1=πxfor all α∈Zk. The code CHis 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∈ Snis defined so that π−1 x(j) = kwhere gk=ggj, it follows that πxfixes no coordinate when x6=α1, and πα1=IdSnfor all α∈Zk. 17 Remark 3.6. A notorious class of cocyclic Butson matrices are those that are equivalent to group invariant (if Gis 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 Gabelian 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 Cpwhen pis prime. Corollary 3.7. Let CHbe a BHFP-code of length nover Zkcoming from H∈BH(n, k), where His row and column balanced. Then His cocyclic. Proof. The proof follows the proof of Proposition 4 and Corollary 2 of [1]. Let Hbe a BH(n, k). We consider the following partition of its corresponding code. CH=∪1≤α≤nCαwhere Cα={[L(H)]α+λ1}λ∈Zkand [L(D)]i denotes the i-th row of L(D). Example 3.8. Let Hbe 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 Πof permutations πx=                        Ix∈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 (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 CHis a BHFP-code with group structure Z8×Z4and Π∼ =Z8. The codewords are C1={[0,0,0,0,0,0,0,0] + λ1}, C2={[0,1,3,0,2,3,1,2] + λ1}, C3={[0,3,2,1,0,3,2,1] + λ1}, C4={[0,0,1,1,2,2,3,3] + λ1}, C5={[0,2,0,2,0,2,0,2] + λ1}, C6={[0,3,3,2,2,1,1,0] + λ1}, C7={[0,1,2,3,0,1,2,3] + λ1}, C8={[0,2,1,3,2,0,3,1] + λ1} where λruns through Z4, and CHis a (8,32,4)-code over Z4.CHhas 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 Zpsare 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 G=Zts p×Zts−1 p2×... ×Zt2 ps−1×Zt1−1 ps, 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 Ht1,t2under 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 Πof permutations Π∼ =Z2×Z4 generated by πxand πywhere 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=x4i. 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 Cobtained from H, a BH(n, mps), is the same as the number of codewords in the BH-code C′ 19 obtained from HΨ. 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 Cobtained from the trivial BH(1,9), and the Z3- code Ψ(C)obtained from the BH(3,3) matrix H′= (1)Ψwhich written in log form is L(H′) =   0 0 0 0 1 2 0 2 1   The code Cis 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∼ =Z9generated by any single element x, no matter what the coordinate permutation πxmay 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+yfor all x,y∈Ψ(C)), and Ψis 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 Ψwill 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 Z2 3. However, we find that for the special case Ψ2:Z4m→Z2 2m, we can carefully construct an isomorphism between the groups of codewords Cand C′= Ψ2(C), and determine the group operation ⋆′so that (C, ⋆)∼ =(C′, ⋆′). Let Ψ = Ψ2. Theorem 4.1. Let mbe an odd positive integer, and let C⊆Zn 4mbe a full propelinear code. Then the code C′= Ψ(C)is full propelinear with group structure (C′, ⋆′)∼ =(C, ⋆). Proof. First observe that Ψis a bijection from Cto 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∈ S2for each x∈Z4mso that Ψ(x) + ρx(Ψ(y)) = Ψ(x+y) for all y. We will see that ρx= (1,2)x, i.e., ρxpermutes the two coordinates of a word in Z2 2mor 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+dtaken 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,xis 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 Cis full propelinear of length n, and let x,y∈C, with x⋆y=x+πx(y). Let πΦ(x)∈ S2npermute the nblocks of size 2, labelled b1,...,bn, according to the action of πxon a word of length n. That is, πΦ(x)(bi) = bjif and only if πx(i) = j. Then πΦ(x)(Ψ(y)) = Ψ(πx(y)). Further, let ρi= (2i−1,2i)be the permutation swapping the entries of the block bi, and write ρx=Qn i=1 ρxi i. It follows that Ψ(x)⋆′Ψ(y) := Ψ(x) + ρxπΦ(x)(Ψ(y)) = Ψ(x+πx(y)) = Ψ(x⋆y). Thus Ψis a bijective homomorphism from (C, ⋆)to (C′, ⋆′). It remains to verify that the permutation ρxπΦ(x)=IdS2nwhenever Ψ(x) = α12nfor any α∈Z2m, and has no fixed coordinate otherwise. Let S=C∩ {α1n: 0 ≤α≤4m−1}and let X⊂Sbe 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 ρxπΦ(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 Zn 4m, and it follows that πΦ(c)does not fix any coordinate of Z2n 2m. Corollary 4.2. Let mbe an odd positive integer, and let H∈BH(n, 4m). If the BH-code Cobtained from His full propelinear with group structure G, then the BH-code C′obtained from HΨwhere Ψis full propelinear with group structure G′∼ =G. Example 4.3. Let H3,0be the BH-code associated to F4⊗F4∈BH(16,4) and H3,0be its image by the Gray map which is known to be a nonlinear code (see [9, Table 1]). H3,0is full propelinear, with permutation group Π∼ =Z2 4 generated by πxand πywhere 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 ρxπΦ(x)and ρyπΦ(y)are as follows: ρxπΦ(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), ρyπΦ(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, H3,0can be endowed with a full propelinear structure with the group hρxπΦ(x), ρyπΦ(y)iof permutations, which is non-abelian of order 32. This group contains the element (ρxπΦ(x))(ρyπΦ(y))(ρxπΦ(x))−1(ρyπΦ(y))−1= ρ1πΦ(1)= (1,2)(3,4) ···(31,32). The groups (H3,0, ⋆)∼ =(H3,0, ⋆′)are isomorphic to Z2×Z4×Z8. Remark 4.4. Even though the codes Cand 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 Z4code and its image in Z2 2. 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 Z4is necessarily trivial, and the group of coordinate permutations of the image is generated by ρ[1]π[0,1] = (1,2). More generally, if Cis a BHFP-code obtained from a BH(n, 4m)with group Πof 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). 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