In this work, we consider pseudocodewords of (relaxed) linear programming
(LP) decoding of 3-dimensional turbo codes (3D-TCs). We present a relaxed LP
decoder for 3D-TCs, adapting the relaxed LP decoder for conventional turbo
codes proposed by Feldman in his thesis. We show that the 3D-TC polytope is
proper and C-symmetric, and make a connection to finite graph covers of the
3D-TC factor graph. This connection is used to show that the support set of any
pseudocodeword is a stopping set of iterative decoding of 3D-TCs using maximum
a posteriori constituent decoders on the binary erasure channel. Furthermore,
we compute ensemble-average pseudoweight enumerators of 3D-TCs and perform a
finite-length minimum pseudoweight analysis for small cover degrees. Also, an
explicit description of the fundamental cone of the 3D-TC polytope is given.
Finally, we present an extensive numerical study of small-to-medium block
length 3D-TCs, which shows that 1) typically (i.e., in most cases) when the
minimum distance dminβ and/or the stopping distance hminβ is
high, the minimum pseudoweight (on the additive white Gaussian noise channel)
is strictly smaller than both the dminβ and the hminβ, and 2)
the minimum pseudoweight grows with the block length, at least for
small-to-medium block lengths.Comment: To appear in IEEE Transactions on Communication