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.

中文翻译：

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素域上非二进制线性码的自适应线性规划解码

在这项工作中，我们考虑了质数域上线性码的自适应线性规划 (ALP) 解码，即有限域 $\mathbb F_{p}$ 大小 $p$ 在哪里 $p$ 是素数，当在 a 上使用时 $p$ -ary 输入无记忆通道。特别是，我们为在任何素数域上的单个奇偶校验 (SPC) 代码的所谓恒权嵌入的码字多面体（或凸包）提供了有效不等式的一般构造（不使用辅助变量） . 该构造基于向量集，称为积木类，根据几条规则组装成不等式的左侧。如果是几乎双对称 有效的类我们证明由此产生的不等式都是 定义面，而我们推测这是真的当且仅当该类有效并且 对称的. 有效的对称类对来自该类的向量的元素强加了某些对称条件，而有效的双对称类则强加了进一步的技术对称条件。为了 $p=3$ ，只有一个有效的对称类，我们证明了由此产生的不等式以及所谓的 单纯形约束给出了嵌入 SPC 代码的码字多胞体的完整和冗余描述。为了 $p > 5$ ，我们表明除了提议的结构之外还有其他方面。例如，对于 $p=7$ ，我们提供了额外的不等式，这些不等式都定义了嵌入的码字多面体的方面。由此产生的一组线性（不）等式被推测为是冗余和完整的。这样的线性（不）等式集合以前没有出现在文献中，具有很强的理论兴趣，我们使用它们开发了一个高效的（宽松的）ALP 解码器，用于一般的（非 SPC）质数域上的线性代码。关键要素是基于动态规划原理的高效分离算法。此外，我们为任意域上的线性代码构建了一个解码器 $\mathbb F_{q}$ 和 $q =p^{m}$ 和 $m>1$ 通过因子图表示，减少到案例的几个实例 $m=1$ ，这通常会导致原始解码多面体的松弛。最后，我们提出了一种有效的切割生成算法来搜索冗余奇偶校验，以进一步提高中短块长度的最大似然解码性能。数值实验证实，对于各种字段大小、校验节点度和块长度，我们的新解码器与静态 LP 解码器相比非常有效。