Bounds on coding gain versus decoding delay for spherical codes on the Gaussian channel

Shannon´s (1949) well-known bound on the capacity of the AWGN channel provides a lower bound on the required signal to noise ratio (SNR) on the channel to achieve a given information rate. It is also well-known, however, that this corresponds to codes of arbitrary length, which implies unlimited decoding delay. In many applications, and especially telephony, delay is severely limited, and may indeed constrain the system performance more fundamentally than decoder complexity. Delay is most fundamentally limited by the length of a code required to achieve the required SNR at the target bit error ratio (BER). (This neglects processing delay). Hence we use bounds on the BER of a block code of given length. The results, nevertheless, are applicable also to convolutional codes. Many such bounds are known, but we use upper and lower bounds appropriate to the class of spherical group codes on the AWGN channel. This class includes all linear binary codes and coded modulation schemes based on MPSK. The lower bound is due to Aldis (1992), but the upper bound does not seem to have previously appeared in the literature