Self-dual skew codes and factorization of skew polynomials

The construction of cyclic codes can be generalized to so-called “module θ-codes” using noncommutative polynomials. The product of the generator polynomial g of a self-dual “module θ  -code” and its “skew reciprocal polynomial” is known to be a noncommutative polynomial of the form Xn−aXn−a, reducing the problem of the computation of all such codes to the resolution of a polynomial system where the unknowns are the coefficients of g. We show that a   must be ±1 and that over F4F4 for n=2sn=2s the factorization of the generator g of a self-dual θ-cyclic code has some rigidity properties which explains the small number of self-dual θ  -cyclic codes with length n=2sn=2s. In the case θ   of order two, we present a construction of self-dual codes, based on the least common multiples of noncommutative polynomials, that allows to reduce the computation to polynomial systems of smaller sizes than the original one. We use this approach to construct a [78,39,19]4[78,39,19]4 self-dual code and a [52,26,17]9[52,26,17]9 self-dual code which improve the best previously known minimal distances for these lengths.