Asymptotically Good Codes Over Non-Abelian Groups

TL;DR

This paper constructs codes over the smallest non-Abelian group, demonstrating their superior performance over Abelian group codes of the same size, which could benefit multi-terminal communication systems.

Contribution

It introduces the first known codes over the smallest non-Abelian group $ ext{D}_6$ with improved performance over Abelian codes, highlighting the potential of non-Abelian group codes.

Findings

Codes over $ ext{D}_6$ outperform Abelian group codes of same size

Non-Abelian codes can be effectively used in multi-terminal settings

Performance gains suggest new directions for code design

Abstract

It has been shown that good structured codes over non-Abelian groups do exist. Specifically, we construct codes over the smallest non-Abelian group $\mathds{D}_6$ and show that the performance of these codes is superior to the performance of Abelian group codes of the same alphabet size. This promises the possibility of using non-Abelian codes for multi-terminal settings where the structure of the code can be exploited to gain performance.