LP Decoding

Linear programming (LP) relaxation is a common technique used to find good solutions to complex optimization problems. We present the method of “LP decoding”: applying LP relaxation to the problem of maximum-likelihood (ML) decoding. An arbitrary binary-input memoryless channel is considered. This treatment of the LP decoding method places our previous work on turbo codes [6] and low-density parity-check (LDPC) codes [8] into a generic framework. We define the notion of a proper relaxation, and show that any LP decoder that uses a proper relaxation exhibits many useful properties. We describe the notion of pseudocodewords under LP decoding, unifying many known characterizations for specific codes and channels. The fractional distance of an LP decoder is defined, and it is shown that LP decoders correct a number of errors equal to half the fractional distance. We also discuss the application of LP decoding to binary linear codes. We define the notion of a relaxation being symmetric for a binary linear code. We show that if a relaxation is symmetric, one may assume that the all-zeros codeword is transmitted.