Self-dual skew codes and factorization of skew polynomials
Résumé
In previous work the authors generalized cyclic codes to the noncommutative polynomial setting and used this approach to construct new self-dual codes over F4. According to this previous result, such a self-dual code must be $\theta$-constacyclic, i.e. the generator polynomial is a right divisor of some noncommutative polynomial $X^n-a$. The first result of the paper is that such a self-dual code must be $\theta$-cyclic or $\theta$-negacyclic, i.e. $a=\pm 1$. For codes of length $2^s$ the noncommutative polynomial approach produced surprisingly poor results. We give an explanation of the length $2^s$ phenomena by showing that in this case the generating skew polynomial has some unique factorization properties. We also construct self-dual skew codes using least common left multiples of noncommutative polynomials and use this to obtain a new $[78,39,19]_4$ self-dual code.
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