2012/11/22 by Ángel del Rı́o, del Rio, Ángel, Josep Rifà +1 · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1211.5251
openalex publication_date 2012/11/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A Z2Z4Q8-code is a non-empty subgroup of a direct product of copies of Z2, Z4 and Q8 (the binary field, the ring of integers modulo 4 and the quaternion group on eight elements, respectively). Such Z2Z4Q8-codes are translation invariant propelinear codes as the well known Z4-linear or Z2Z4-linear codes. In the current paper, we show that there exist "pure" Z2Z4Q8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the Z2Z4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of Z2Z4Q8-codes; we study and show the different shapes of such a codes and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard.