Minimum Pseudoweight Analysis of 3-Dimensional Turbo Codes

TL;DR

This paper analyzes the pseudoweights of 3D turbo codes under LP decoding, providing theoretical insights, enumerators, and numerical results that relate pseudoweight to code parameters and block length.

Contribution

It introduces a relaxed LP decoder for 3D-TCs, characterizes the fundamental cone, and connects pseudocodewords to stopping sets, advancing understanding of decoding performance.

Findings

Minimum pseudoweight is often smaller than minimum distance and stopping distance.

Minimum pseudoweight increases with block length for small-to-medium sizes.

The fundamental cone of 3D-TCs is explicitly described.

Abstract

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 $d_{\rm min}$ and/or the stopping distance $h_{\rm min}$ is high, the minimum pseudoweight (on the additive white Gaussian noise channel) is strictly smaller than both the $d_{\rm min}$ and the $h_{\rm min}$, and 2) the minimum pseudoweight grows with the block length, at least for small-to-medium block lengths.