On the Connection Between Finite Graph Covers, Pseudo-Codewords, and Linear Programming Decoding of Turbo Codes

The performance of turbo decoding on the binary erasure channel (BEC) can be characterized in terms of turbo stopping sets. Apply turbo decoding until the transmitted codeword has been recovered, or the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum-size turbo stopping set which is also a subset of the set of erased positions. The concept of turbo stopping sets is an adaptation of stopping sets from the theory of iterative belief-propagation (BP) decoding of low-density parity-check (LDPC) codes. In this work we consider relaxed linear programming (LP) decoding of turbo codes as described by Feldman in his thesis. We show that the set of points from the relaxed polytope where all entries are rational numbers is equal to the set of all pseudo-codewords of all unite graph covers of the turbo code factor graph. We also show that there is a many-to-one correspondence between pseudo-codewords and turbo stopping sets in the following sense. The support set of any pseudo-codeword, i.e., the set of non-zero coordinates, is a turbo stopping set, and for any turbo stopping set there is a pseudo-codeword with support set equal to the turbo stopping set. Furthermore, the minimum additive white Gaussian noise (AWGN) pseudo-weight of pseudo-codewords with support set equal to a minimal type-1 turbo stopping set is equal to its support set size. Finally, we present some simulation results showing the infuence of small-size turbo stopping sets on turbo decoding on the AWGN channel.