Coding with skew polynomial rings

• Autores: D. Boucher, Felix Ulmer
• Localización: Journal of symbolic computation, ISSN 0747-7171, Vol. 44, Nº 12, 2009, págs. 1644-1656
• Idioma: inglés
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• Resumen

  • In analogy to cyclic codes, we study linear codes over finite fields obtained from left ideals in a quotient ring of a (non-commutative) skew polynomial ring. The paper shows how existence and properties of such codes are linked to arithmetic properties of skew polynomials. This class of codes is a generalization of the O-cyclic codes discussed in [Boucher, D., Geiselmann, W., Ulmer, F., 2007. Skew cyclic codes. Applied Algebra in Engineering, Communication and Computing 18, 379�389]. However O-cyclic codes are powerful representatives of this family and we show that the dual of a O-cyclic code is still O-cyclic. Using Groebner bases, we compute all Euclidean and Hermitian self-dual O-cyclic codes over F4 of length less than 40, including a [36,18,11] Euclidean self-dual O-cyclic code which improves the previously best known self-dual code of length 36 over F4.