Description
This work is focused on the structure and nature of the decoding failures of low-density parity-check (LDPC) codes, namely, pseudocodewords. These (pseudocodewords) may come from either linear programming (LP) decoding or message-passing iterative decoding (MPID) algorithms for LDPC codes. Since the former decoding algorithm is more amenable to analysis, our focus is on pseudocodewords associated with it. There is no loss of generality because it can be shown that the two algorithms are equivalent. We state the theorem and proof showing that the perm-vector associated to a code where the components are the permanents of sub-matrices of a parity-check matrix is in fact a pseudocodeword. The calculation of the permanent of a matrix is known to be in the complexity class #P, that is, the set of the counting problems associated with the decision problems in the class NP, hence the interest in the Bethe permanent as introduced by Vontobel. Smarandache proves that: 1) vectors whose components are the Bethe-permanent of certain sub-matrices and 1) degree-M Bethe perm-vectors are both pseudocodewords. This work seeks to be a self-contained introduction to the subject, from the precise definition of pseudocodewords to examples involving their creation from permanent and Bethe perm-vectors. Future work would require precisely identifying how these pseudocodewords may contribute as a bound for the fundamental cone.
