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TitleSelf-dual skew codes and factorization of skew polynomials
Author(s) Delphine Boucher, Felix Ulmer
TypeArticle in Journal
AbstractAbstract 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 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 self-dual θ-cyclic code has some rigidity properties which explains the small number of self-dual θ-cyclic codes with length n = 2^s . 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 self-dual code and a [ 52 , 26 , 17 ] 9 self-dual code which improve the best previously known minimal distances for these lengths.
KeywordsError-correcting codes, Finite fields, Skew polynomial rings
ISSN0747-7171
URL http://www.sciencedirect.com/science/article/pii/S0747717113001223
LanguageEnglish
JournalJournal of Symbolic Computation
Volume60
Number0
Pages47 - 61
Year2014
Edition0
Translation No
Refereed No
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