We Can Think of Good Codes, and Even Decode Them
“All codes are good, except those we can think of.” This pessimistic opinion is widely shared among the coding theory community. We shall try instead to defend the optimistic statement of the above title. The word “good” is given in the quoted sentence the restricted meaning: “with a large minimum distance”. From a communication engineering point of view, we were led to criticize this definition and to propose as a better criterion of goodness the proximity of the normalized distance distribution of a code with respect to that which results in the average from random coding, as measured for instance by the cross-entropy of these distributions. Looking for codes which are good in this sense for the Euclidean metric, as relevant to the additive white Gaussian noise channel, we show that combining a maximum distance separable code (e.g., a Reed-Solomon one) with an almost arbitrary one-to-one mapping of its q-ary symbols into a 2-dimensional constellation is a satisfactory solution provided q is large enough.
Near-capacity performance may be expected from such a random-like code, provided it can be decoded with reasonably low complexity. Using the Euclidean metric implies soft-decision decoding. We discuss two decoding algorithms for this purpose. The first one is sequential, as adapted to the finite context of a block code. It is potentially optimum, although practical considerations lead to limit the number of words to be tried. The second one is intrinsically nonoptimum and relies on an interpretation of a linear code as a kind of product of single-check codes which enables their cascaded weighted-output decoding. For large Reed-Solomon codes, none of these algorithms is fully satisfactory, but we suggest to combine them in order to obtain a near-optimum algorithm of acceptable complexity.
KeywordsLinear Code Distance Distribution Random Code Information Symbol Additive White Gaussian Noise Channel
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