Introduction to Linear Optimization
Introduction to Linear Optimization
Probabilistic analysis of linear programming decoding
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Modern Coding Theory
IEEE Transactions on Information Theory - Part 1
Factor graphs and the sum-product algorithm
IEEE Transactions on Information Theory
The capacity of low-density parity-check codes under message-passing decoding
IEEE Transactions on Information Theory
Error exponents of expander codes
IEEE Transactions on Information Theory
On decoding of low-density parity-check codes over the binary erasure channel
IEEE Transactions on Information Theory
Extrinsic information transfer functions: model and erasure channel properties
IEEE Transactions on Information Theory
Using linear programming to Decode Binary linear codes
IEEE Transactions on Information Theory
MAP estimation via agreement on trees: message-passing and linear programming
IEEE Transactions on Information Theory
LP Decoding Corrects a Constant Fraction of Errors
IEEE Transactions on Information Theory
Iterative reliability-based decoding of low-density parity check codes
IEEE Journal on Selected Areas in Communications
Nonlinear programming approaches to decoding low-density parity-check codes
IEEE Journal on Selected Areas in Communications
Iterative approximate linear programming decoding of LDPC codes with linear complexity
IEEE Transactions on Information Theory
A separation algorithm for improved LP-decoding of linear block codes
IEEE Transactions on Information Theory
| Hi-index | 754.96 |
We investigate the structure of the polytope underlying the linear programming (LP) decoder introduced by Feldman, Karger, and Wainwright. We first show that for expander codes, every fractional pseudocodeword always has at least a constant fraction of nonintegral bits. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than that of any codeword. We further exploit these geometrical properties to devise an improved decoding algorithm with the same order of complexity as LP decoding that provably performs better. The method is very simple: it first applies, ordinary LP decoding, and when it fails, it proceeds by guessing facets of the polytope, and then resolving the linear program on these facets. While the LP decoder succeeds only if the ML codeword has the highest likelihood over all pseudocodewords, we prove that the proposed algorithm, when applied to suitable expander codes, succeeds unless there exists a certain number of pseudocodewords, all adjacent to the ML codeword on the LP decoding polytope, and with higher likelihood than the ML codeword. We then describe an extended algorithm, still with polynomial complexity, that succeeds as long as there are at most polynomially many pseudocode words above the ML codeword.