2020/10/13 by José Ándrés Armario, Armario, José Andrés, Iván Bailera +3 · 1 citation
Engineering · Computer Science · #graph theory and CDMA systems #Coding theory and cryptography
paper · pdf · doi:10.48550/arxiv.2010.06206
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 k\rm th roots of unity, we can construct a larger Butson matrix over the ℓ\rm th 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.