Z2Z4-Additive Cyclic Codes: Kernel and Rank

Abstract

A Z 2 Z 4 -additive code C ⊆ Z α 2 × Z β 4 is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z 2 coordinates and the set of Z 4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Φ(C) be the binary Gray map image of C. We study the rank and the dimension of the kernel of a Z 2 Z 4 -additive cyclic code C, that is, the dimensions of the binary linear codes Φ(C) and ker(Φ(C)). We give upper and lower bounds for these parameters. It is known that the codes Φ(C) and ker(Φ(C)) are binary images of Z 2 Z 4 -additive codes R(C) and K(C), respectively. Moreover, we show that R(C) and K(C) are also cyclic and we determine the generator polynomials of these codes in terms of the generator polynomials of the code C.Keywords Z 2 Z 4 -additive cyclic codes, Gray map, kernel, rank. I. INTRODUCTIONDenote by Z 2 and Z 4 the rings of integers modulo 2 and modulo 4, respectively. We denote the space of n-tuples over these rings as Z n 2 and Z n 4 . A binary code is any non-empty subset C

$\mathbb{Z}_2\mathbb{Z}_4$-Additive Cyclic Codes Are Asymptotically Good

“…[10] studied the generator matrices and dual codes of Z 2 Z 4 -additive cyclic codes. Borges, Dougherty, Fernández-Córdoba and Ten-Valls study the rank and the kernel of Z 2 Z 4 -additive cyclic codes in [8].…”