Codes from unit groups of division algebras over number fields

TL;DR

This paper demonstrates that codes constructed from unit groups of division algebras over number fields are asymptotically good for the sum-rank metric, expanding the scope of algebraic code constructions.

Contribution

It proves that noncommutative unit group constructions produce asymptotically good codes from division algebras of any degree, with estimates on alphabet size.

Findings

Codes from division algebra unit groups are asymptotically good.

The construction applies to division algebras of any degree.

Alphabet size is estimated in relation to the algebra degree.

Abstract

Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, the first author and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. We prove that the noncommutative unit group construction yields asymptotically good families of codes for the sum-rank metric from division algebras of any degree, and we estimate the size of the alphabet in terms of the degree.