Decoding error-correcting codes via linear programming
Decoding error-correcting codes via linear programming
IEEE Transactions on Information Theory
An efficient algorithm to find all small-size stopping sets of low-density parity-check matrices
IEEE Transactions on Information Theory
Good error-correcting codes based on very sparse matrices
IEEE Transactions on Information Theory
Finite-length analysis of low-density parity-check codes on the binary erasure channel
IEEE Transactions on Information Theory
Using linear programming to Decode Binary linear codes
IEEE Transactions on Information Theory
On the stopping distance and the stopping redundancy of codes
IEEE Transactions on Information Theory
Pseudocodewords of Tanner Graphs
IEEE Transactions on Information Theory
An Efficient Pseudocodeword Search Algorithm for Linear Programming Decoding of LDPC Codes
IEEE Transactions on Information Theory
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In this work, we consider the pairwise error probability (PEP) of a linear programming (LP) decoder for a general binary linear code as formulated by Feldman et al. (IEEE Trans. Inf. Theory, March 2005) on a quantized additive white Gaussian noise (AWGN) channel. With a quantized AWGN (QAWGN) channel, we mean a channel where we first compute log-likelihood ratios as for an AWGN channel and then quantize them. Let H be a parity-check matrix of a binary linear code and consider LP decoding based on H. The output of the LP decoder is always a pseudo-codeword , of some pseudo-weight , where the definition of pseudo-weight is specific to the underlying channel model. In this work, we give a definition of pseudo-weight for a QAWGN channel based on an asymptotic (high signal-to-noise ratio) analysis of the PEP. Note that with maximum-likelihood decoding, the parameters of the quantization scheme, i.e., the quantization levels and the corresponding quantization region thresholds, that minimize the PEP of wrongly decoding to a non-zero codeword c when the all-zero codeword is transmitted is independent of the specific codeword c. However, this is not the case with LP decoding based on a parity-check matrix H, which means that the quantization scheme needs to be optimized for the given H. As a case study, we consider the well-known (3,5)-regular (155,64,20) Tanner code and estimate its minimum QAWGN pseudo-weight with 3 and 5 levels of quantization, in which the quantization scheme is optimized to maximize the minimum QAWGN pseudo-weight.