Z2-Triple cyclic codes and their duals

TL;DR

This paper investigates Z2-triple cyclic codes, characterizing their structure, generators, and duals, providing a comprehensive algebraic framework for understanding these codes in polynomial form.

Contribution

It determines the structure, generators, and duals of Z2-triple cyclic codes, extending the algebraic understanding of these codes.

Findings

Explicit form of generators for Z2-triple cyclic codes

Minimal generating sets identified

Dual code structures characterized

Abstract

A Z2-triple cyclic code of block length (r,s,t) is a binary code of length r+s+t such that the code is partitioned into three parts of lengthsr,s andt such that each of the three parts is invariant under the cyclic shifts of the coordinates. Such a code can be viewed as Z2[x]-submodules of Z_2[x]/<x^r-1>xZ_2[x]/<x^s-1>xZ_2[x]/<x^t-1>, in polynomial representation. In this paper, we determine the structure of these codes. We have obtained the form of the generators for such codes. Further, a minimal generating set for such a code is obtained. Also, we study the structure of the duals of these codes via the generators of the codes.