Generalized Hadamard full propelinear codes

Butson full propelinear codes

“…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 than any linear code of the same size, or by having more codewords than any linear code with a given minimum distance [1,13]. In this paper we extend the work of the authors in [1] and describe the connection between cocyclic Butson Hadamard matrices and BHFP-codes.…”

“…In Sect. 3 we construct an explicit morphism BH(n, k) → BH(nm, k/m) where k = p e 1 1 • • • p e t t and m = p e 1 −1 study. The rows of any H ∈ BH(n, k) scaled by a factor of 1/ √ n is an orthonormal basis of C n .…”

“…The code C H over Z k is called a Butson Hadamard code (briefly, BH-code). [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]…”

“…In the binary case, we have the Hadamard full propelinear codes, they were introduced in [26] and their equivalence with Hadamard groups was proven. In the q-ary case, 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)-difference sets.…”

q-ary Propelinear Perfect Codes from the Regular Subgroups of the GA(r,q) and Their Ranks

“…Unlike binary codes, which are well-studied, there are rather few works devoted to q-ary propelinear codes, for q ≥ 3. We refer to [7], [2], [8], [1] for q-ary perfect, MDS and generalized Hadamard propelinear codes.…”