[Paper Review] Self-dual cyclic codes over finite chain rings

Bocong Chen, San Ling+1 more|arXiv (Cornell University)|May 11, 2014

Coding theory and cryptography21 references3 citations

TL;DR

This paper investigates the algebraic structure of self-dual cyclic codes over finite chain rings, establishing that such codes exist if and only if the nilpotency index $ t $ of the ring's maximal ideal is even. It provides explicit necessary and sufficient conditions for existence and derives an enumeration formula based on the number of self-reciprocal irreducible factors of $ X^n - 1 $ over the residue field $ \mathbb{F}_q $, with closed-form expressions for $ n $ having two prime divisors.

ABSTRACT

Let $R$ be a finite commutative chain ring with unique maximal ideal $\langle γ angle$, and let $n$ be a positive integer coprime with the characteristic of $R/\langle γ angle$. In this paper, the algebraic structure of cyclic codes of length $n$ over $R$ is investigated. Some new necessary and sufficient conditions for the existence of nontrivial self-dual cyclic codes are provided. An enumeration formula for the self-dual cyclic codes is also studied.

Motivation & Objective

• To characterize the algebraic structure of cyclic codes of length $ n $ over finite chain rings $ R $, where $ n $ is coprime to the characteristic of the residue field $ \mathbb{F}_q = R/\langle\gamma\rangle $.
• To determine necessary and sufficient conditions for the existence of nontrivial self-dual cyclic codes over such rings.
• To derive an enumeration formula for the number of self-dual cyclic codes, linking it to the number of self-reciprocal monic irreducible factors of $ X^n - 1 $ over $ \mathbb{F}_q $.
• To compute $ |\Omega_n| $, the number of self-reciprocal monic irreducible factors of $ X^n - 1 $, particularly for $ n $ with exactly two distinct prime divisors.

Proposed method

• Generalizes methods from [13] to derive a new algebraic structure for cyclic codes of length $ n $ over $ R $, differing from prior work in [5].
• Uses the ring homomorphism $ R[X] \to \mathbb{F}_q[X] $ to lift properties of polynomials over $ \mathbb{F}_q $ to $ R $, leveraging the correspondence between coprime polynomials in $ R[X] $ and $ \mathbb{F}_q[X] $.
• Analyzes the monic irreducible factorization of $ X^n - 1 $ over $ \mathbb{F}_q $, focusing on reciprocal and self-reciprocal factors to determine conditions for self-dual codes.
• Reduces the computation of $ |\Omega_{2^m n'}| $ to $ |\Omega_{n'}| $ and $ |\overline{\Omega}_{n'}| $, where $ \overline{\Omega}_{n'} $ counts self-reciprocal factors over $ \mathbb{F}_{q^2} $.
• Applies results on orders of $ q $ modulo $ \ell_i^{r_i} $ to classify cases based on $ a_i = v_2(\text{ord}_{\ell_i^{r_i}}(q)) $, leading to case distinctions in the enumeration formula.
• Derives explicit formulas for $ |\Omega_n| $ when $ n $ has exactly two prime divisors, distinguishing cases based on equality of $ a_1 $ and $ a_2 $.

Experimental results

Research questions

• RQ1 Under what conditions do nontrivial self-dual cyclic codes of length $ n $ exist over a finite chain ring $ R $?
• RQ2 How does the nilpotency index $ t $ of the maximal ideal $ \langle\gamma\rangle $ affect the existence of self-dual cyclic codes?
• RQ3 What is the exact number of self-dual cyclic codes of length $ n $ over $ R $, and how is it determined by the factorization of $ X^n - 1 $ over $ \mathbb{F}_q $?
• RQ4 How can $ |\Omega_n| $, the number of self-reciprocal monic irreducible factors of $ X^n - 1 $ over $ \mathbb{F}_q $, be computed efficiently for $ n $ with two prime divisors?
• RQ5 What role does the structure of $ \text{ord}_d(q) $ play in classifying the number of self-dual cyclic codes?

Key findings

• Self-dual cyclic codes of length $ n $ over a finite chain ring $ R $ exist if and only if the nilpotency index $ t $ of the maximal ideal $ \langle\gamma\rangle $ is even.
• When $ t $ is even, the number of self-dual cyclic codes is completely determined by $ |\Delta_n| $, the number of reciprocal polynomial pairs in the monic irreducible factorization of $ X^n - 1 $ over $ \mathbb{F}_q $.
• The problem of computing $ |\Omega_n| $, the number of self-reciprocal monic irreducible factors of $ X^n - 1 $ over $ \mathbb{F}_q $, is reduced to computing $ |\Omega_{n'}| $ and $ |\overline{\Omega}_{n'}| $, where $ n = 2^m n' $, $ n' $ odd.
• For $ n = \ell_1^{r_1} \ell_2^{r_2} $ with distinct odd primes $ \ell_1, \ell_2 $, the value of $ |\Omega_n| $ is explicitly computed in two subcases: when $ a_1 = a_2 \geq 1 $, it is $ \sum_{d \mid n} \frac{\phi(d)}{\text{ord}_d(q)} $; when $ a_1 \neq a_2 $, it is $ \sum_{d_1=0}^{r_1} \frac{\phi(\ell_1^{d_1})}{\text{ord}_{\ell_1^{d_1}}(q)} + \sum_{d_2=0}^{r_2} \frac{\phi(\ell_2^{d_2})}{\text{ord}_{\ell_2^{d_2}}(q)} - 1 $.
• When $ a_1 = 0 $ or $ a_2 = 0 $, the formula simplifies: $ |\Omega_n| = |\Omega_{\ell_2^{r_2}}| $ or $ |\Omega_n| = |\Omega_{\ell_1^{r_1}}| $, respectively.