Details:
Title  Selfdual skew codes and factorization of skew polynomials  Author(s)  Delphine Boucher, Felix Ulmer  Type  Article in Journal  Abstract  Abstract The construction of cyclic codes can be generalized to socalled “module θcodes” using noncommutative polynomials. The product of the generator polynomial g of a selfdual “module θcode” and its “skew reciprocal polynomial” is known to be a noncommutative polynomial of the form X^n − 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 F 4 for n = 2^s the factorization of the generator g of a selfdual θcyclic code has some rigidity properties which explains the small number of selfdual θcyclic codes with length n = 2^s . In the case θ of order two, we present a construction of selfdual 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 selfdual code and a [ 52 , 26 , 17 ] 9 selfdual code which improve the best previously known minimal distances for these lengths.  Keywords  Errorcorrecting codes, Finite fields, Skew polynomial rings  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717113001223 
Language  English  Journal  Journal of Symbolic Computation  Volume  60  Number  0  Pages  47  61  Year  2014  Edition  0  Translation 
No  Refereed 
No 
