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.
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