A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates linear-programming based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the 'maximum-likelihood certificate' property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. Two alternative polytopes for use with linear-programming decoding are studied, and it is shown that for many classes of codes these polytopes yield a complexity advantage for decoding. These polytope representations lead to polynomial-time decoders for a wide variety of classical nonbinary linear codes. LP decoding performance is illustrated for the [11,6] ternary Golay code with ternary PSK modulation over AWGN, and in this case it is shown that the performance of the LP decoder is comparable to codeword-error-rate-optimum hard-decision based decoding. LP decoding is also simulated for medium-length ternary and quaternary LDPC codes with corresponding PSK modulations over AWGN.Comment: To appear in the IEEE Transactions on Information Theory. 54 pages, 5 figure

F. J. MacWilliams proved that Hamming isometries between linear codes extend to monomial transformations. This theorem has recently been generalized by J. Wood who proved it for Frobenius rings using character theoretic methods. The present paper provides a combinatorial approach: First we extend I. Constantinescu's concept of homogeneous weights on arbitrary finite rings and prove MacWilliams' equivalence theorem to hold with respect to these weights for all finite Frobenius rings. As a central tool we then establish a general inversion principle for real functions on finite modules that involves Mo bius inversion on partially ordered sets. An application of the latter yields the aforementioned result of Wood. Academic Press

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The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting.

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