Decoding in weighted combinatorial and other metrics

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There exist low-complexity hard-input bounded distance decoders for many block codes. However, low complexity soft-input decoding remains still a challenging task. Our goal is to propose a soft-input decoding of block codes for channels with memory. For memoryless channels such soft-input decoding, called Generalized Minimum Distance (GMD) decoding, was suggested by Forney in 1966. GMD decoding uses reliabilities of received symbols and can be considered as a bounded distance decoding in a weighted (by the reliabilities) Hamming metric. The Hamming metric matches discrete memoryless symmetric channels, i.e., maximum-likelihood decoding in such channels is equivalent to minimum distance decoding in the Hamming metric. GMD decoding uses a hardinput error-and-erasure decoder of the block code in a multitrial manner, where in each decoding trial, a number of least reliable received symbols are erased before decoding. Hundreds of publications show that GMD decoding is a universal procedure (applicable to an arbitrary code) with very good performance and low complexity. As a result, it has many practical applications for different memoryless channels and for the decoding (generalized) concatenated codes.