Impact of redundant parity checks on linear programming decoding of LDPC codes -

Abstract

The use of linear programming (LP) to decode low density parity check codes was first suggested by Feldman[1]. It is a relaxation of the maximum likelihood decoding problem into a linear optimization problem of polynomial complexity. The underlying polytope is obtained by adding all the local linear constraints associated with the individual check nodes in the Tanner graph of the code. This relaxation results in invalid codewords being included in the polytope of the LP solver. While it had been suggested by Feldman that the size of the polytope may be reduced through the addition of redundant parity checks – parity checks obtained through the combination of other parity checks – the issue was not further examined in that paper. The purpose of this thesis is to carry out such an analysis and determine what reduction in polytope size may be possible through the addition of these redundant checks. It will be shown that in the case of good girth LDPC codes, and assuming reasonable conjectures on the fundamental polytope of the LP decoder to be true, the LDPC codes typically encountered in practice, such redundant checks do not improve the error correction capability of these codes under LP decoding.