Trinucleotide k-circular codes I: theory
Résumé
A code X is (≥ k)-circular if any concatenation of at most k words from X, when read on a circle, admits exactly one partition into words from X. A code that is (≥ k)-circular for all integers k is said to be circular. Any code is (≥ 0)-circular and it turns out that a code of trinucleotides is circular as soon as it is (≥ 4)-circular. A code is k-circular if it is (≥ k)-circular and not (≥ k+1)-circular. Due to the explosive combinatorics of trinucleotide k-circular codes, we developed three classes of algorithms based on: (i) the smallest directed cycles (directed girth) in graphs; (ii) the eigenvalues of matrices; and (iii) the files that incrementally save partial results. These different approaches also allow us to verify the computational results obtained. We determine here the growth functions of trinucleotide k-circular codes, k varying between 0 and 4, in the general case and in various particular cases: minimum, minimal, maximum, self-complementary, (k, k, k)-circular and self-complementary (k, k, k)-circular.
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