Z2-double cyclic codes

A binary linear code $C$ is a $\mathbb{Z}_2$-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the $\mathbb{Z}_2[x]$-module $\mathbb{Z}_2[x]/(x^r-1)\times\mathbb{Z}_2[x]/(x^s-1).$ We determine the structure of $\mathbb{Z}_2$-double cyclic codes giving the generator polynomials of these codes. The related polynomial representation of $\mathbb{Z}_2$-double cyclic codes and its duals, and the relations between the polynomial generators of these codes are studied.

💡 Research Summary

This paper introduces and thoroughly analyzes a new family of binary linear codes termed Z2-double cyclic codes. A binary linear code C of length n = r + s is defined as a Z2-double cyclic code if, for any codeword, applying a cyclic shift independently within the first r coordinates and within the last s coordinates results in another codeword. This structure generalizes classical cyclic codes (where one subset is empty) and is permutation-equivalent to quasi-cyclic codes of index 2 when r = s.

The core of the paper establishes a robust algebraic framework for these codes. The authors demonstrate a fundamental isomorphism: Z2-double cyclic codes can be precisely identified as submodules of the Z2