{'fraction_non_alphanumeric': 0.02631578947368421, 'fraction_numerical': 0.0, 'mean_word_length': 4.928571428571429, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Constructing valid convex hull inequalities for single parity-check codes over prime fields

In this work, we present an explicit construction of valid inequalities (using no auxiliary variables) for 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 classes of building blocks 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. Such sets of inequalities have not appeared in the literature before, have a strong theoretical interest, and can be used to develop an efficient (relaxed) adaptive linear programming decoder for general (non-SPC) linear codes over prime fields.