On the Effective Weights of Pseudocodewords for Codes Defined on Graphs with Cycles

The behavior of an iterative decoding algorithm for a code defined on a graph with cycles and a given decoding schedule is characterized by a cycle-free computation tree. The pseudocodewords of such a tree are the words that satisfy all tree constraintsj pseudocodewords govern decoding performance. Wiberg [12] determined the effective weight of pseudocodewords for binary codewords on an AWGN channel. This paper extends Wiberg’s formula for AWGN channels to nonbinary codes, develops similar results for BSC and BEC channels, and gives upper and lower bounds on the effective weight. The 16-state tail-biting trellis of the Golay code [2] is used for exampIes. Although in this case no pseudocodeword is found with effective weight less than the minimum Hamming weight of the Golay code on an AWGN channel, it is shown by example that the minimum effective pseudocodeword weight can be less than the minimum codeword weight.