[Paper Review] Construction and enumeration for self-dual cyclic codes of even length over $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$

Yuan Cao, Yonglin Cao+3 more|arXiv (Cornell University)|Jul 14, 2019

Coding theory and cryptography15 references4 citations

TL;DR

This paper provides a complete algebraic construction and enumeration of self-dual cyclic codes of even length $2^s n$ over the finite chain ring $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$ with $u^2 = 0$, using a recursive lifting method based on trace inverses and factorization of $x^N - 1$. The key contribution is a closed-form formula for counting all such self-dual codes, enabling the systematic generation of self-dual and $2$-quasi-cyclic codes over $\mathbb{F}_{2^m}$ via the Gray map.

ABSTRACT

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $R=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ and $s,n$ be positive integers such that $n$ is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring $R$ of length $2^sn$ and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and $2$-quasi-cyclic codes over $\mathbb{F}_{2^m}$ of length $2^{s+1}n$ can be obtained from self-dual cyclic code over $R$ of length $2^sn$ and by a Gray map preserving orthogonality and distances from $R$ onto $\mathbb{F}_{2^m}^2$.

Motivation & Objective

• To provide a complete classification and explicit construction of self-dual cyclic codes of even length over the finite chain ring $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$.
• To derive a closed-form formula for the number of distinct self-dual cyclic codes of length $2^s n$ where $n$ is odd.
• To establish a systematic method for generating self-dual and $2$-quasi-cyclic codes over $\mathbb{F}_{2^m}$ from self-dual cyclic codes over $R = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$ via the Gray map.
• To generalize prior results on constacyclic and cyclic codes over $\mathbb{F}_2 + u\mathbb{F}_2$ to the more general ring $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$ for even-length codes.

Proposed method

• The method uses a recursive lifting approach based on the factorization of $x^N - 1$ into irreducible factors over $\mathbb{F}_{2^m}$, particularly focusing on the case $N = 2^s n$ with $n$ odd.
• It introduces a trace inverse construction: $\mathcal{W}^{(2,s;k)}_{(\beta_0,\dots,\beta_{k-1})} = x^{-(4+k)} \cdot \operatorname{Tr}^{-1}(x^{4+k} \cdot \delta^{(k)}_{(\beta_0,\dots,\beta_{k-1})}(x))$, where $\delta^{(k)}$ are polynomials satisfying certain congruences modulo powers of $f_2(x)$.
• The construction proceeds step-by-step: for each $k$, the set $\mathcal{W}^{(2,s;k)}_{(\beta_0,\dots,\beta_{k-1})}$ is computed using the previous level's values and solving a system of congruences involving $f_2(x)^k$.
• The method relies on the structure of the ring $R = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$ and the isomorphism $\phi: R \to \mathbb{F}_{2^m}^2$, which preserves duality and weight distributions under the Gray map.
• The algorithm computes all solutions recursively by solving for $\delta^{(k)}$ polynomials that satisfy the required congruence relations modulo $f_2(x)^k$, ensuring the self-dual condition is preserved.
• The final code is expressed as $\Omega_{2,s} = \sum_{k=0}^{3} \beta_k(x) f_2(x)^k$, with $\beta_k(x)$ drawn from precomputed sets $\mathcal{W}^{(2,s;k)}$.

Experimental results

Research questions

• RQ1 How can self-dual cyclic codes of even length $2^s n$ over $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$ be explicitly constructed?
• RQ2 What is the exact number of distinct self-dual cyclic codes of length $2^s n$ over $R = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$?
• RQ3 How can the Gray map be used to lift self-dual codes over $R$ to self-dual and $2$-quasi-cyclic codes over $\mathbb{F}_{2^m}$?
• RQ4 What algebraic structure enables the recursive construction of such codes via trace inverse and polynomial congruences?
• RQ5 Can the method be generalized to other even-length cyclic codes over finite chain rings of the form $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$?

Key findings

• The paper provides a complete and explicit construction of all self-dual cyclic codes of length $2^s n$ over $R = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$, where $n$ is odd, using a recursive trace inverse method.
• A closed-form formula for the number of such self-dual cyclic codes is derived, based on the number of solutions to a system of polynomial congruences modulo powers of $f_2(x)$.
• For the case $s=2$, $m=1$, and $n=1$, the construction yields $\Omega_{2,4}$ with $16$ elements: $8$ from the $\beta_0 = 0$ branch and $8$ from the $\beta_0 = 1+x$ branch.
• The Gray map $\phi$ preserves duality and weight distribution, so every self-dual cyclic code over $R$ of length $N$ maps to a self-dual $2$-quasi-cyclic code over $\mathbb{F}_{2^m}$ of length $2N$.
• The construction is effective and systematic: for each level $k$, the set $\mathcal{W}^{(2,s;k)}$ is computed recursively using trace inverse and polynomial lifting, ensuring all solutions are captured.
• The method generalizes previous results on constacyclic and cyclic codes over $\mathbb{F}_2 + u\mathbb{F}_2$ to the larger ring $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$, extending applicability to codes over larger finite fields.