Diletter and triletter comma-free codes over finite alphabets
Résumé
Comma-free codes have been widely studied in the last sixty years, from points of view as diverse as biology, information theory and combinatorics. We develop new methods to study comma-free codes achieving the maximum size, given the cardinality of the alphabet and the length of the words. Specifically, we are interested in counting the number of such codes when all words have length 2, or 3. We first explain how different properties combine to obtain a closed-formula. We next develop an approach to tackle well-known sub-families of comma-free codes, such as self-complementary and (generalisations of) non-overlapping codes, for which the aforementioned properties do not hold anymore. We also study codes that are not contained in strictly larger ones. For instance, we determine the maximal size of self-complementary comma-free codes (over an alphabet of arbitrary cardinality) and the number of codes reaching the bound. We also provide a characterisation of non-overlapping triletter codes that are inclusion-wise maximal, which allows us to devise the number of such codes. We point out other applications of the method, notably to self-complementary codes, including the recently introduced mixed codes. Our approaches mix combinatorial and graph-theoretical arguments.
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