[Paper Review] Rank weight hierarchy of some classes of cyclic codes

Jérôme Ducoat, Frédérique Oggier|arXiv (Cornell University)|Jul 15, 2014

Coding theory and cryptography2 references4 citations

TL;DR

This paper establishes the rank weight hierarchy of cyclic codes over finite fields $\mathbb{F}_{q^m}$, deriving exact formulas for $[n,n-1]$ cyclic codes and characterizing codes with rank metric 1 under specific conditions. It introduces a refined Singleton bound for cyclic codes when $n$ and $q$ are coprime, significantly advancing the understanding of rank metric properties in network coding applications.

ABSTRACT

We study the rank weight hierarchy, thus in particular the rank metric, of cyclic codes over the finite field $\mathbb F_{q^m}$, $q$ a prime power, $m \geq 2$. We establish the rank weight hierarchy for $[n,n-1]$ cyclic codes and characterize $[n,k]$ cyclic codes of rank metric 1 when (1) $k=1$, (2) $n$ and $q$ are coprime, and (3) the characteristic $char(\mathbb F_q)$ divides $n$. Finally, for $n$ and $q$ coprime, cyclic codes of minimal $r$-rank are characterized, and a refinement of the Singleton bound for the rank weight is derived.

Motivation & Objective

• To determine the complete rank weight hierarchy for $[n,n-1]$ cyclic codes over $\mathbb{F}_{q^m}$.
• To characterize $[n,k]$ cyclic codes with rank metric 1 when $k=1$, $\gcd(n,q)=1$, or $\mathrm{char}(\mathbb{F}_q) \mid n$.
• To identify cyclic codes of minimal $r$-rank and derive a refined Singleton bound for the rank weight under the condition that $n$ and $q$ are coprime.
• To extend the theoretical framework of rank metric codes for network coding applications, particularly in wiretap scenarios.

Proposed method

• Uses the matrix representation $\lambda(c)$ of codewords via an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^m}$ to define rank weight as the $\mathbb{F}_q$-rank of the matrix.
• Applies the definition of rank weight hierarchy via subspaces $V \in \Gamma(\mathbb{F}_{q^m}^n)$ satisfying $V^q = V$, minimizing $\dim V$ under $\dim(C \cap V) \geq r$.
• Employs the dual code $C^\perp$ and the duality relation $\{d_r(\lambda(C))\} \sqcup \{n+1 - d_s(\lambda(C^\perp))\} = \{1,\dots,n\}$ to derive bounds.
• Leverages the $q$-degree $\eta_q(C)$ of a code, defined as the sum of degrees of minimal polynomials of roots of the generator polynomial over $\mathbb{F}_q$, to characterize rank properties.
• Applies the discrete Fourier transform and polynomial algebra techniques to analyze codewords and their $\mathbb{F}_q$-linear spans.
• Derives a refined Singleton bound: $d(\lambda(C)) \leq \min(\eta_q(C^\perp) - k + 1, m)$ for $\gcd(n,q)=1$.

Experimental results

Research questions

• RQ1 What is the exact rank weight hierarchy for $[n,n-1]$ cyclic codes over $\mathbb{F}_{q^m}$?
• RQ2 Under what conditions do $[n,k]$ cyclic codes have rank metric 1, particularly when $k=1$, $\gcd(n,q)=1$, or $\mathrm{char}(\mathbb{F}_q) \mid n$?
• RQ3 How can cyclic codes of minimal $r$-rank be characterized when $n$ and $q$ are coprime?
• RQ4 Can a tighter bound than the standard Singleton bound be derived for the rank weight of cyclic codes under coprime $n$ and $q$?
• RQ5 What is the relationship between the $q$-degree $\eta_q(C)$ and the rank weight hierarchy of cyclic codes?

Key findings

• The rank weight hierarchy of $[n,n-1]$ cyclic codes is fully determined: $d_r(\lambda(C)) = r$ for all $1 \leq r \leq n-1$, meaning the code is $r$-MRD for all $r$.
• For $[n,1]$ cyclic codes, the minimum rank distance is always 1, regardless of $n$ and $q$, confirming earlier results in the literature.
• When $k=1$, $\gcd(n,q)=1$, or $\mathrm{char}(\mathbb{F}_q) \mid n$, the paper provides a complete characterization of $[n,k]$ cyclic codes with rank metric 1.
• For $\gcd(n,q)=1$, the paper proves that $d_r(\lambda(C)) = r$ if and only if $\eta_q(C) \leq n - r$, linking the $q$-degree to the rank weight hierarchy.
• A refined Singleton bound is derived: $d(\lambda(C)) \leq \min(\eta_q(C^\perp) - k + 1, m)$, which improves upon the classical bound when $\gcd(n,q)=1$.
• An example shows that the refined bound can be tight: for a $[11,8]$ code over $\mathbb{F}_{3^5}$, $\eta_q(C^\perp) = 10$, yielding $d(\lambda(C)) \leq 3$, and the actual minimum rank distance is 2.