[Paper Review] On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic

Ping Li, Wei Dai+1 more|arXiv (Cornell University)|Nov 10, 2016

Coding theory and cryptography1 references3 citations

TL;DR

This paper investigates $ζ_2\u03b6_2[u]$-$(1+u)$-additive constacyclic codes of arbitrary length, establishing their algebraic structure as submodules of a polynomial ring $R_{\alpha,\beta}$. It derives minimal generating sets, characterizes the dual code via generator polynomials, and analyzes Gray images, providing explicit formulas for code parameters based on generator degrees.

ABSTRACT

In this paper, we study $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic code of arbitrary length. Firstly, we study the algebraic structure of this family of codes and a set of generator polynomials for this family as a $(\mathbb{Z}_{2}+u\mathbb{Z}_{2})[x]$-submodule of the ring $R_{α,β}$. Secondly, we give the minimal generating sets of this family codes, and we determine the relationship of generators between the $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic codes and its dual and give the parameters in terms of the degrees of the generator polynomials of the code. Lastly, we also study $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic code in terms of the Gray images.

Motivation & Objective

• To characterize the algebraic structure of $ζ_2\u03b6_2[u]$-$(1+u)$-additive constacyclic codes as $R[x]$-submodules of $R_{\alpha,\beta}$.
• To determine minimal generating sets for these codes and establish relationships between generators of the code and its dual.
• To derive explicit formulas for code parameters in terms of the degrees of generator polynomials.
• To analyze the Gray image of these codes and its relation to the dual code under the Gray map.
• To extend the understanding of constacyclic codes over mixed alphabets in the context of $ζ_2\u03b6_2[u]$-additive codes.

Proposed method

• Represent codewords as pairs of polynomials $(a(x), b(x))$ in the ring $R_{\alpha,\beta} = \mathbb{Z}_2[x]/\langle x^\alpha - 1 \rangle \times R[x]/\langle x^\beta - 1 - u \rangle$, identifying the code as an $R[x]$-submodule.
• Define the $(1+u)$-constacyclic shift and show that the code is closed under this operation, establishing its constacyclic nature.
• Use the Gray map $\phi$ to transform the code into a binary linear code of length $\alpha + 2\beta$, preserving distance properties.
• Derive the dual code's generator polynomials using the inner product and polynomial duality, particularly involving $\gcd$ and reciprocal polynomials.
• Establish that $\phi(C^\perp) = \phi(C)^\perp$, linking the dual of the original code to the dual of its binary image.
• Apply results from polynomial rings over $\mathbb{Z}_2$ to express dual generators in terms of $\gcd(a(x), l(x))^*$ and reciprocal polynomials.

Experimental results

Research questions

• RQ1 How can the algebraic structure of $ζ_2\u03b6_2[u]$-$(1+u)$-additive constacyclic codes be characterized as submodules of $R_{\alpha,\beta}$?
• RQ2 What are the minimal generating sets for these codes, and how do they relate to the code's parameters?
• RQ3 How can the generators of the dual code be expressed in terms of the generators of the original code?
• RQ4 What is the relationship between the Gray image of the code and the dual code?
• RQ5 How do the degrees of the generator polynomials determine the code's parameters?

Key findings

• The code $C = \langle (a(x), 0), (l(x), g(x)) \rangle$ is fully characterized by its generator polynomials in $R_{\alpha,\beta}$, with $a(x)$ dividing $x^\alpha - 1$ and $g(x)$ dividing $x^\beta - 1 - u$.
• The dual code $C^\perp$ has generators $\bar{a}(x) = \frac{x^\alpha - 1}{\gcd(a(x), l(x))^*}$, $\bar{l}(x) = \frac{x^\alpha - 1}{a^*(x)} \lambda(x)$, and $\bar{g}(x) = \frac{(x^{2\beta} - 1) \gcd(a(x), l(x))^*}{a^*(x) g^*(x)}$, where $\lambda(x) \in \mathbb{Z}_2[x]$.
• The Gray image $\phi(C)$ is a binary linear code of length $\alpha + 2\beta$, and $\phi(C^\perp) = \phi(C)^\perp$, preserving duality under the map.
• The dimension of $C$ is $k_1 + 2k_2$, with $|C| = 2^{k_1} 4^{k_2}$, and the dual has size $|C^\perp| = 2^{\alpha + k_1 - 2k_0} 4^{\beta - k_1 - k_2 + k_0}$, where $k_0$ is the dimension of the subcode $C_b$.
• For codes of the form $C = \langle (a(x), 0), (l(x), u g(x)) \rangle$, the dual generators are derived similarly, with $\bar{g}(x)$ involving $x^{2\beta} - 1$ and reciprocal polynomials.
• When $g(x) = f_1^{i_1}(x) \cdots f_r^{i_r}(x)$, the dual generator $\bar{f}_j(x)$ is given by $\frac{(x^{2\beta} - 1) \gcd(a(x), l(x))^*}{a^*(x) f_j^{*i_j}(x)}$, showing a multiplicative duality structure.