Cyclic codes over $M_2(\F_2)$

TL;DR

This paper explores cyclic codes over the non-commutative ring M_2(F_2), deriving structure theorems and characterizing lengths for self-dual codes, with applications to quaternionic lattices and space-time coding.

Contribution

It provides the first structural analysis of cyclic codes over M_2(F_2) and characterizes conditions for self-duality, advancing coding theory over non-commutative rings.

Findings

Derived structure theorems for cyclic codes over M_2(F_2)

Characterized lengths where self-dual cyclic codes exist

Constructed formally self-dual quaternary codes

Abstract

The ring in the title is the first non commutative ring to have been used as alphabet for block codes. The original motivation was the construction of some quaternionic modular lattices from codes. The new application is the construction of space time codes obtained by concatenation from the Golden code. In this article, we derive structure theorems for cyclic codes over that ring, and use them to characterize the lengths where self dual cyclic codes exist. These codes in turn give rise to formally self dual quaternary codes.