[Paper Review] On the Lattice of Cyclic Linear Codes Over Finite Chain Rings

Alexandre Fotue Tabue, Christophe Mouaha|arXiv (Cornell University)|Jan 30, 2017

Coding theory and cryptography3 references3 citations

TL;DR

This paper develops a trace representation for free cyclic linear codes over finite chain rings using $q$-cyclotomic cosets when $\gcd(\ell, q) = 1$, and investigates the lattice structure $\langle \text{Cy}(R,\ell); +, \cap \rangle$ of such codes. It establishes a lower bound on the Hamming distance and constructs MDS and self-orthogonal codes when $q$ is even, extending classical results from finite fields to chain rings.

ABSTRACT

Let $ exttt{R}$ be a commutative finite chain ring of invariants $(q,s).$ In this paper, the trace representation of any free cyclic $ exttt{R}$-linear code of length $\ell,$ is presented, via the $q$-cyclotomic cosets modulo $\ell,$ when $ exttt{gcd}(\ell, q) = 1.$ The lattice $\left( exttt{Cy}( exttt{R},\ell), +, \cap ight)$ of cyclic $ exttt{R}$-linear codes of length $\ell,$ is investigated. A lower bound on the Hamming distance of cyclic $ exttt{R}$-linear codes of length $\ell,$ is established. When $q$ is even, a family of MDS and self-orthogonal $ exttt{R}$-linear cyclic codes, is constructed.

Motivation & Objective

• To extend the theory of cyclic linear codes from finite fields to finite chain rings, particularly under the condition $\gcd(\ell, q) = 1$.
• To develop a trace-based representation for free cyclic $R$-linear codes using $q$-cyclotomic cosets.
• To investigate the algebraic lattice structure $\langle \text{Cy}(R,\ell); +, \cap \rangle$ of cyclic codes over finite chain rings.
• To establish a lower bound on the minimum Hamming distance of cyclic $R$-linear codes.
• To construct families of MDS and self-orthogonal cyclic codes when $q$ is even.

Proposed method

• Utilizes the polynomial representation $\Psi: \mathbb{R}^\ell \to \mathbb{R}[X]/(X^\ell - 1)$ to identify cyclic codes with ideals in the quotient ring.
• Employs $q$-cyclotomic cosets modulo $\ell$ to parameterize and describe the trace representation of free cyclic codes over finite chain rings.
• Applies the trace map over Galois extensions of finite chain rings to construct codewords from trace images of elements in extension rings.
• Derives the lattice operations $+$ and $\cap$ on cyclic codes via the correspondence with ideals in $\mathbb{R}[X]/(X^\ell - 1)$, using $\gcd$ and $\text{lcm}$ of generator polynomials.
• Uses the structure of the chain ring $R$ with maximal ideal $\mathfrak{J}(R)$ and residue field $\mathbb{F}_q$ to lift properties from $\mathbb{F}_q$ to $R$.
• Constructs MDS and self-orthogonal codes by analyzing the duals of trace-generated codes and exploiting the symmetry in cyclotomic cosets when $q$ is even.

Experimental results

Research questions

• RQ1 How can the trace representation of free cyclic linear codes over finite chain rings be systematically constructed using $q$-cyclotomic cosets when $\gcd(\ell, q) = 1$?
• RQ2 What is the structure of the lattice $\langle \text{Cy}(R,\ell); +, \cap \rangle$ of cyclic $R$-linear codes, and how does it relate to the ideal structure of $\mathbb{R}[X]/(X^\ell - 1)$?
• RQ3 Can a lower bound on the minimum Hamming distance of cyclic $R$-linear codes be established based on their trace representation?
• RQ4 Under what conditions can MDS or self-orthogonal cyclic codes be constructed over finite chain rings, particularly when $q$ is even?
• RQ5 How do the duals of trace-generated cyclic codes behave, and what symmetries arise in their lattice structure?

Key findings

• A trace representation of any free cyclic $R$-linear code of length $\ell$ is established via $q$-cyclotomic cosets when $\gcd(\ell, q) = 1$, enabling systematic construction.
• The lattice $\langle \text{Cy}(R,\ell); +, \cap \rangle$ is shown to be distributive under the given conditions, generalizing the field case.
• A lower bound on the minimum Hamming distance of cyclic $R$-linear codes is derived based on the number of $q$-cyclotomic cosets in the generator set.
• When $q$ is even, the paper constructs families of MDS and self-orthogonal cyclic codes over finite chain rings using trace-based generator sets.
• For $\mathbb{Z}_4$-linear codes of length 7, the paper explicitly lists 27 cyclic codes with their types, cardinalities, and dualities, including $\mathcal{C}_8^\perp = \mathcal{C}_{19}$ and $\mathcal{C}_8 + \mathcal{C}_{12} = \mathcal{C}_{15}$, demonstrating the lattice structure concretely.
• The duality and sum relations in the lattice, such as $\mathcal{C}_8 \cap \mathcal{C}_{12} = \mathcal{C}_6$, are verified via duals and the orthogonal complement operation.