On the non-Abelian group code capacity of memoryless channels

Abstract

In this work is provided a definition of group encoding capacity C G of non-Abelian group codes transmitted through symmetric channels. It is shown that this C G is an upper bound of the set of rates of these non-Abelian group codes that allow reliable transmission. Also, is inferred that the C G is a lower bound of the channel capacity. After that, is computed the C G of the group code over the dihedral group transmitted through the 8PSK-AWGN channel then is shown that it equals the channel capacity. It remains an open problem whether there exist non-Abelian group codes of rate arbitrarily close to C G and arbitrarily small error probability.

On Symmetric Channels and Codes Over the Quaternion Group

“…On the other hand it may be the case that a channel to be symmetric for more than one group. For instance the 8PSK-AWGN channel is G-symmetric for both G = Z 8 (cyclic) and G = D 4 [4], [3]. A final observation on Definition 2 is that the G-Symmetric channels preserve the two important properties of classical symmetric channels:…”

“…From this, U ≈ C. For the specific case of the quaternion group, a group code must be a subgroup of (Z 4 ⊠ Z 2 ) N . A useful formula, shown in [4], is…”

“…It is an extension of [3] where group-capacity was investigated for channels over Abelian groups. Also, it can be considered as a continuation of [4] where it was shown analytically that C D4 , the group-capacity for the dihedral group case, equals the channel capacity. The main difference between both the dihedral and quaternion cases comes from the dimension of the respective channels.…”