On Coding Without Restrictions

Many coded modulation constructions, such as lattice codes, are visualized as restricted subsets of an infinite cmstellation (IC) of points in the n-dimensional Euclidean space. We shall regard an IC as a code without restrictions employed for the AWGN channel. For an IC the concept of coding rate is meaningless and we shall use, instead of coding rate, the normalized logarithmic density (NLD). The maximum value Csuch that, for any NLD less than COO, it is possible to construct an IC with arbitrarily small decoding error probability, will be called the generalized capacity of the AWGN channel without restrictions. We derive exponential upper and lower bounds for the decoding error probability of an IC, expressed in terms of the NLD. The upper bound is obtained by means of a random coding method and it is very similar to the usual random coding bound for the AWGN channel. The exponents of these upper and lower bounds coincide for high values of the NLD, thereby enabling derivation of the generalized capacity of the AWGN channel without restrictions. It is also shown that the exponent of the random coding bound can be attained by linear IC’s (lattices), implying that lattices play the same role with respect to the AWGN channel as linear codes do with respect to a discrete symmetric channel.