Z2Z4-Additive Cyclic Codes: Kernel and Rank

TL;DR

This paper investigates the properties of Z2Z4-additive cyclic codes, focusing on their binary Gray images, and provides bounds and generator polynomial characterizations for their rank and kernel dimensions.

Contribution

It introduces bounds for the rank and kernel of Z2Z4-additive cyclic codes and characterizes the generator polynomials of related cyclic codes R(C) and K(C).

Findings

Bounds for the rank and kernel dimensions are established.

R(C) and K(C) are cyclic codes with determined generator polynomials.

Binary images of these codes are characterized in terms of their generator polynomials.

Abstract

A Z2Z4-additive code C subset of Z_2^alpha x Z_4^beta is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z_2 and the set of Z_4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Phi(C) be the binary Gray image of C. We study the rank and the dimension of the kernel of a Z2Z4-additive cyclic code C, that is, the dimensions of the binary linear codes <Phi(C)> and ker(Phi(C)). We give upper and lower bounds for these parameters. It is known that the codes <Phi(C)> and ker(Phi(C)) are binary images of Z2Z4-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.