Coding for the Gaussian Multiple-Access Channel: An Algebraic Approach

Given any group G, the G-adder channel is the channel with T inputs taking values in G and output equal to the sum (over G) of the inputs. An F-adder channel is a G-adder channel where G is the additive group of a finite field F. Similarly, the R-adder channel is the one corresponding to the usual field R of real numbers. The Gaussian multiple-access channel is the cascade of the R-adder channel with the (single-user) additive white Gaussian noise channel. Multiple-access multiple-rate codes for F-adder channels are defined and two constructions for such codes are given. In order to use such codes on the Gaussian multiple-access channel, the latter is decomposed into a number, say l, of F-adder channels. This is done via a construction involving a lattice with sufficient coding gain to reduce the error-probability to a negligible value and sublattices of it by means of which we form a suitable chain of finite quotient groups. The multiple-access codes described are well suited for use with random-access protocols with multiple reception.