[Paper Review] A note on linear codes and nonassociative algebras obtained from skew-polynomial rings

Susanne Pumpluen|arXiv (Cornell University)|Apr 1, 2015

Coding theory and cryptography1 references3 citations

TL;DR

This paper unifies various constructions of linear codes—such as $σ$-constacyclic, ideal $σ$-codes, and module $σ$-codes—by showing they all arise from left ideals in a nonassociative algebra $S_f$ derived from skew-polynomial rings via Petit's construction. The key result is that a linear code over $\mathbb{F}_q$ of length $m$ is $\sigma$-constacyclic (with constant $d$) if and only if its skew-polynomial representation forms a left ideal in $S_f$ generated by a monic right divisor of $f = t^m - d$ in $\mathbb{F}_q[t;\sigma]$. This provides a unified algebraic framework for code construction using nonassociative structures.

ABSTRACT

Different approaches to construct linear codes using skew polynomials can be unified by using the nonassociative algebras built from skew-polynomial rings by Petit.

Motivation & Objective

• To unify disparate constructions of linear codes from skew-polynomial rings into a single algebraic framework.
• To show that various types of $\sigma$-codes (constacyclic, ideal, module) are all instances of left ideals in a nonassociative algebra $S_f$.
• To establish a necessary and sufficient condition for a linear code to be $\sigma$-constacyclic using the skew-polynomial representation and the algebra $S_f$.
• To clarify the role of the two-sidedness of $f = t^m - d$ and its implications on code existence and structure.

Proposed method

• Construct the nonassociative algebra $S_f = R_m$ with multiplication defined by right division modulo $f$, where $R = \mathbb{F}_q[t;\sigma,\delta]$ and $\deg(f) = m$.
• Use Petit's construction to define a nonassociative algebra structure on the set of polynomials of degree less than $m$ using the right-division operation $g \circ h = gh \mod_r f$.
• Characterize left ideals in $S_f$ as being generated by right divisors of $f$, linking them directly to code structures.
• Apply the theory of skew-polynomial rings and their left and right division algorithms to ensure existence and uniqueness of representations.
• Use the skew-polynomial representation of code vectors as elements in $\mathbb{F}_q[t;\sigma]/(f)$ to analyze closure under left multiplication by elements of $S_f$.
• Establish equivalence between $\sigma$-constacyclic codes and left ideals in $S_f$ via the condition that $f = t^m - d$ is a two-sided element when $d \in \mathrm{Fix}(\sigma)$ and $m$ divides the order of $\sigma$.

Experimental results

Research questions

• RQ1 Can different constructions of $\sigma$-codes (e.g., constacyclic, ideal, module) be unified under a single algebraic structure?
• RQ2 What is the precise algebraic condition under which a linear code over $\mathbb{F}_q$ of length $m$ is $\sigma$-constacyclic?
• RQ3 How does the nonassociative algebra $S_f$ constructed from skew-polynomial rings relate to the structure of $\sigma$-codes?
• RQ4 Under what conditions is $f = t^m - d$ a two-sided element in $\mathbb{F}_q[t;\sigma]$, and how does this affect code construction?
• RQ5 When is $f = t^m - d$ irreducible in $\mathbb{F}_q[t;\sigma]$, and what are the implications for the existence of nontrivial $\sigma$-codes?

Key findings

• A linear code $\mathcal{C}$ over $\mathbb{F}_q$ of length $m$ is $\sigma$-constacyclic with constant $d$ if and only if its skew-polynomial representation $\mathcal{C}(t)$ is a left ideal in the nonassociative algebra $S_f$ generated by a monic right divisor of $f = t^m - d$ in $\mathbb{F}_q[t;\sigma]$.
• The construction of $S_f$ via Petit's method provides a unified framework: all $\sigma$-codes (constacyclic, ideal, module) arise as left ideals in $S_f$ when $f$ is reducible.
• The two-sidedness of $f = t^m - d$ is equivalent to $m$ dividing the order of $\sigma$ and $d \in \mathrm{Fix}(\sigma)$, which ensures $S_f$ is associative and $f$ is reducible.
• For $m$ prime and $\mathrm{Fix}(\sigma)$ containing a primitive $m$th root of unity, $f = t^m - d$ is reducible in $\mathbb{F}_q[t;\sigma]$ if and only if $d$ is a norm-like product of $\sigma$-images of some $z \in \mathbb{F}_q$.
• If $f = t^m - d$ is irreducible in $\mathbb{F}_q[t;\sigma]$, then $S_f$ is a finite division algebra (a semifield), and the only $\sigma$-constacyclic codes are the trivial $[m,m]$-code and the full space.
• The nonassociative algebra $S_f$ is isomorphic to a cyclic algebra $(K/F, \sigma, d)$ when $K/F$ is a Galois extension of degree $m$ and $f = t^m - d$ with $d \notin F$, linking code theory to finite semifields and spreads.