On skew polynomial codes and lattices from quotients of cyclic division algebras

TL;DR

This paper introduces a new method for constructing lattices from skew polynomial codes derived from cyclic division algebras, with applications to space-time coding.

Contribution

It presents a novel variation of Construction A using skew polynomial codes from quotients of cyclic division algebras, expanding lattice construction techniques.

Findings

Established an isomorphism between quotient rings and skew polynomial rings.

Analyzed properties of skew polynomial codes and their duals.

Applied the construction to space-time coding scenarios.

Abstract

We propose a variation of Construction A of lattices from linear codes defined using the quotient $\Lambda/\mathfrak p\Lambda$ of some order $\Lambda$ inside a cyclic division $F$-algebra, for $\mathfrak p$ a prime ideal of a number field $F$. To obtain codes over this quotient, we first give an isomorphism between $\Lambda/\mathfrak p\Lambda$ and a ring of skew polynomials. We then discuss definitions and basic properties of skew polynomial codes, which are needed for Construction A, but also explore further properties of the dual of such codes. We conclude by providing an application to space-time coding, which is the original motivation to consider cyclic division $F$-algebras as a starting point for this variation of Construction A.