In recent years, several integer programming (IP) approaches were developed for maximum-likelihood decoding and minimum distance computation for binary linear codes. Two aspects in particular have been demonstrated to improve the performance of IP solvers as well as adaptive linear programming decoders: the dynamic generation of forbidden-set (FS) inequalities, a family of valid cutting planes, and the utilization of so-called redundant parity-checks (RPCs). However, to date, it had remained unclear how to solve the exact RPC separation problem (i.e., to determine whether or not there exists any violated FS inequality w.r.t. any known or unknown parity-check). In this note, we prove NP-hardness of this problem. Moreover, we formulate an IP model that combines the search for most violated FS cuts with the generation of RPCs, and report on computational experiments. Empirically, for various instances of the minimum distance problem, it turns out that while utilizing the exact separation IP does not appear to provide a computational advantage, it can apparently be avoided altogether by combining heuristics to generate RPC-based cuts.

中文翻译：

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与二进制线性码的冗余奇偶校验相关的禁止集切割的精确分离

近年来，为二进制线性码的最大似然解码和最小距离计算开发了几种整数规划 (IP) 方法。已经证明了两个方面可以提高 IP 求解器和自适应线性编程解码器的性能：禁用集 (FS) 不等式的动态生成、有效切割平面族以及所谓的冗余奇偶校验的利用-检查（RPC）。然而，迄今为止，仍然不清楚如何解决确切的 RPC 分离问题（即，确定是否存在任何违反任何已知或未知奇偶校验的 FS 不等式）。在这篇笔记中，我们证明了这个问题的 NP-hardness。此外，我们制定了一个 IP 模型，将搜索最违反的 FS 切割与生成 RPC 相结合，并报告计算实验。根据经验，对于最小距离问题的各种实例，事实证明，虽然利用精确分离 IP 似乎没有提供计算优势，但显然可以通过结合启发式方法生成基于 RPC 的切割来完全避免这种情况。