The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Factoring in skew-polynomial rings over finite fields
Journal of Symbolic Computation
On circulant self-dual codes over small fields
Designs, Codes and Cryptography
Coding with skew polynomial rings
Journal of Symbolic Computation
Codes as Modules over Skew Polynomial Rings
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
Composition collisions and projective polynomials: statement of results
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
A note on the dual codes of module skew codes
IMACC'11 Proceedings of the 13th IMA international conference on Cryptography and Coding
On Self-Dual Cyclic Codes Over Finite Fields
IEEE Transactions on Information Theory
| Hi-index | 0.00 |
The construction of cyclic codes can be generalized to so-called ''module @q-codes'' using noncommutative polynomials. The product of the generator polynomial g of a self-dual ''module @q-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 @q-cyclic code has some rigidity properties which explains the small number of self-dual @q-cyclic codes with length n=2^s. In the case @q 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.