Free cyclic codes over finite local rings
Bull. Korean Math. Soc. 2006 Vol. 43, No. 4, 723-735
Published online December 1, 2006
Sung Sik Woo
Ewha Women's University
Abstract : In [2] it was shown that a 1-generator quasi-cyclic code $C$ of length $n=ml$ of index $l$ over $\Bbb Z_4$ is free if $C$ is generated by a polynomial which divides $X^m-1$. In this paper, we prove that a necessary and sufficient condition for a cyclic code over $\Bbb Z_{p^k}$ of length $m$ to be free is that it is generated by a polynomial which divides $X^m-1$. We also show that this can be extended to finite local rings with a principal maximal ideal.
Keywords : free modules over a finite commutative rings, separable extension of local rings, cyclic codes over $\Bbb Z_{p^k}$
MSC numbers : 13C10
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