Reflection Group Codes and Their Decoding

This paper builds on Mittelholzer and Lahtonen's study of group codes for the Gaussian channel based on reflection groups. A careful analysis of the action of a reflection group on its roots leads to the development of improved methods for encoding and decoding. The new algorithm is proved to achieve maximum likelihood decoding. The complexity of decoding is analyzed, and it is shown that a proper choice of the sequence of subgroups used in the algorithm can yield significant gains in the efficiency of decoding.