Codes over non-Abelian groups: Point-to-point communications and computation over MAC

In this paper, we show that good structured codes over non-Abelian groups do exist. Specifically, we construct codes over the smallest non-Abelian group D₆ 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. We also show that for the problem of computation over MAC, these codes are superior to random codes in certain cases.