Adaptive Linear Programming Decoding of Nonbinary Linear Codes Over Prime Fields

摘要

In this work, we consider adaptive linear programming (ALP) decoding of linear codes over prime fields, i.e., the finite fields $mathbb F_{p}$ of size $p$ where $p$ is a prime, when used over a $p$ -ary input memoryless channel. In particular, we provide a general construction of valid inequalities (using no auxiliary variables) for the codeword polytope (or the convex hull) of the so-called constant-weight embedding of a single parity-check (SPC) code over any prime field. The construction is based on sets of vectors, called building block classes, that are assembled to form the left-hand side of an inequality according to several rules. In the case of almost doubly-symmetric valid classes we prove that the resulting inequalities are all facet-defining, while we conjecture this to be true if and only if the class is valid and symmetric. Valid symmetric classes impose certain symmetry conditions on the elements of the vectors from the class, while valid doubly-symmetric classes impose further technical symmetry conditions. For $p=3$ , there is only a single valid symmetric class and we prove that the resulting inequalities together with the so-called simplex constraints give a complete and irredundant description of the codeword polytope of the embedded SPC code. For $p > 5$ , we show that there are additional facets beyond those from the proposed construction. As an example, for $p=7$ , we provide additional inequalities that all define facets of the embedded codeword polytope. The resulting overall set of linear (in)equalities is conjectured to be irredundant and complete. Such sets of linear (in)equalities have not appeared in the literature before, have a strong theoretical interest, and we use them to develop an efficient (relaxed) ALP decoder for general (non-SPC) linear codes over prime fields. The key ingredient is an efficient separation algorithm based on the principle of dynamic programming. Furthermore, we construct a decoder for linear codes over arbitrary fields $mathbb F_{q}$ with $q =p{m}$ and $m>1$ by a factor graph representation that reduces to several instances of the case $m=1$ , which results, in general, in a relaxation of the original decoding polytope. Finally, we present an efficient cut-generating algorithm to search for redundant parity-checks to further improve the performance towards maximum-likelihood decoding for short-to-medium block lengths. Numerical experiments confirm that our new decoder is very efficient compared to a static LP decoder for various field sizes, check-node degrees, and block lengths.