SIGNAL SETS MATCHED TO GROUPS

Binary linear codes are well known to be "matched" to binary signaling on a Gaussian channel. Recently, linear codes over Z(M) (the ring of integers mod M) have been presented that are similarly matched to M-ary phase modulation. Motivated by these new codes, the general problem of matching signal sets to generalized linear algebraic codes is addressed. A definition is given for the notion of matching. It is shown that any signal set in N-dimensional Euclidean space that is matched to an abstract group is essentially what Slepian had called a "group code for the Gaussian channel." If the group is commutative, this further implies that any such signal set is equivalent to coded phase modulation with linear codes over Z(M). It is well known that, for high signal-to-noise ratio, phase modulation does not effectively exploit the capacity of the bandlimited Gaussian channel. The mentioned result, however, implies that all signal sets that are matched to commutative groups are subject to these same limits on performance. This motivates the investigation of signal sets matched to noncommutative groups, and of "linear" codes over such groups.