Coding with skew polynomial rings

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 theta-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 theta-cyclic codes are powerful representatives of this family and we show that the dual of a theta-cyclic code is still theta-cyclic. Using Groebner bases, we compute all Euclidean and Hermitian self-dual theta-cyclic codes over F(4) of length less than 40, including a [36, 18, 11] Euclidean self-dual theta-cyclic code which improves the previously best known self-dual code of length 36 over F(4). (c) 2008 Elsevier Ltd. All rights reserved.