Coding with skew polynomial rings

Abstract

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 θ -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 θ -cyclic codes are powerful representatives of this family and we show that the dual of a θ -cyclic code is still θ -cyclic. Using Groebner bases, we compute all Euclidean and Hermitian self-dual θ -cyclic codes over F 4 of length less than 40, including a [ 36 , 18 , 11 ] Euclidean self-dual θ -cyclic code which improves the previously best known self-dual code of length 36 over F 4 .