[Paper Review] On double cyclic codes over Z_4

Jian Gao, Minjia Shi+2 more|arXiv (Cornell University)|Jan 7, 2015

Coding theory and cryptography4 references3 citations

TL;DR

This paper investigates double cyclic codes over ℤ₄, determining their generator polynomials and minimal generating sets as R[x]-submodules of R[x]/(x^r−1) × R[x]/(x^s−1). It establishes the duality relationship between a double cyclic code and its dual, and demonstrates that optimal or suboptimal nonlinear binary codes can be derived via the Gray map, extending constructions from ℤ₄-cyclic codes to a generalized double-cyclic structure.

ABSTRACT

Let $R=\mathbb{Z}_4$ be the integer ring mod $4$. A double cyclic code of length $(r,s)$ over $R$ is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as $R[x]$-submodules of $R[x]/(x^r-1) imes R[x]/(x^s-1)$. In this paper, we determine the generator polynomials of this family of codes as $R[x]$-submodules of $R[x]/(x^r-1) imes R[x]/(x^s-1)$. Further, we also give the minimal generating sets of this family of codes as $R$-submodules of $R[x]/(x^r-1) imes R[x]/(x^s-1)$. Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual.

Motivation & Objective

• To characterize the algebraic structure of double cyclic codes over ℤ₄ as R[x]-submodules of R[x]/(x^r−1) × R[x]/(x^s−1).
• To determine the generator polynomials and minimal generating sets for these codes.
• To establish the duality relationship between a double cyclic code and its dual over ℤ₄.
• To demonstrate that optimal or suboptimal nonlinear binary codes can be constructed from this family of codes via the Gray map.

Proposed method

• Represent double cyclic codes as R[x]-submodules of the product ring R[x]/(x^r−1) × R[x]/(x^s−1), where R = ℤ₄.
• Use polynomial factorization and Hensel lifting to derive generator polynomials F₁(x), l(x), and F₂(x) for the code.
• Apply the Gray map to transform ℤ₄-codes into binary codes and evaluate their nonlinear properties.
• Derive the dual code’s generator polynomials using inverse polynomials and modular arithmetic in the quotient ring.
• Establish the relationship between the generators of a code and its dual using gcd and reciprocal polynomial operations.
• Verify results through explicit examples, including computation of dual code generators and code sizes.

Experimental results

Research questions

• RQ1 How can double cyclic codes over ℤ₄ be characterized algebraically as R[x]-submodules of R[x]/(x^r−1) × R[x]/(x^s−1)?
• RQ2 What are the minimal generating sets for double cyclic codes over ℤ₄, and how do they relate to the generator polynomials?
• RQ3 What is the precise relationship between the generator polynomials of a double cyclic code and those of its dual code over ℤ₄?
• RQ4 Can optimal or suboptimal nonlinear binary codes be constructed from double cyclic codes over ℤ₄ via the Gray map?
• RQ5 How do the degrees and gcds of the generator polynomials influence the size and structure of the dual code?

Key findings

• The paper determines the generator polynomials of double cyclic codes over ℤ₄ as R[x]-submodules of R[x]/(x^r−1) × R[x]/(x^s−1), providing a complete algebraic characterization.
• It derives the minimal generating sets for these codes, showing that they are spanned by specific polynomial combinations over ℤ₄.
• The dual code of a free double cyclic code is also a double cyclic code, and its generators are explicitly determined using gcd and reciprocal polynomial operations.
• The dual code’s generator polynomials are given by F₁*(x) = (x^r−1)/gcd(F₁(x),l(x)) and F₂*(x) = (x^s−1)gcd(F₂(x),l(x))/(F₁(x)F₂(x)) when the code is free.
• An example shows that a (3,9)-code with |C| = 4⁴ generates a dual code of size 4⁸, confirming the duality structure.
• The construction yields optimal or suboptimal nonlinear binary codes via the Gray map, extending known results from ℤ₄-cyclic codes.