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Finding the minimum distance of linear codes is an NP-hard problem.Traditionally, this computation has been addressed by means of the design ofalgorithms that find, by a clever exhaustive search, a linear combination ofsome generating matrix rows that provides a codeword with minimum weight.Therefore, as the dimension of the code or the size of the underlying finitefield increase, so it does exponentially the run time. In this work, we provethat, given a generating matrix, there exists a column permutation which leadsto a reduced row echelon form containing a row whose weight is the codedistance. This result enables the use of permutations as representation scheme,in contrast to the usual discrete representation, which makes the search of theoptimum polynomial time dependent from the base field. In particular, we haveimplemented genetic and CHC algorithms using this representation as a proof ofconcept. Experimental results have been carried out employing codes over fieldswith two and eight elements, which suggests that evolutionary algorithms withour proposed permutation encoding are competitive with regard to existingmethods in the literature. As a by-product, we have found and amended someinaccuracies in the MAGMA Computational Algebra System concerning the storeddistances of some linear codes.

Genetic algorithms with permutation-based representation for computing the distance of linear codes