How to obtain lattices from (f,sigma,delta)-codes via a generalization of Construction A

TL;DR

This paper introduces a novel method to derive lattices from cyclic (f,σ,δ)-codes over finite rings using quotients of orders in nonassociative division algebras, generalizing classical Construction A and prior associative algebra approaches.

Contribution

It extends lattice construction techniques to nonassociative division algebras via (f,σ,δ)-codes, broadening the scope of algebraic coding theory.

Findings

Generalizes Construction A for lattices from linear codes

Includes previous associative algebra results as special cases

Potential applications in coset and wire-tap coding

Abstract

We show how cyclic $(f,\sigma,\delta)$-codes over finite rings canonically induce a $\mathbb{Z}$-lattice in $\mathbb{R}^N$ by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial $f$. This construction generalizes the one using certain $\sigma$-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by $f$, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap coding. Previous results by Ducoat and Oggier are obtained as special cases.