# FREE CYCLIC CODES OVER FINITE LOCAL RINGS

• Published : 2006.11.30
• 51 5

#### Abstract

In [2] it was shown that a 1-generator quasi-cyclic code C of length n = ml of index l over $\mathbb{Z}_4$ 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 $\mathbb{Z}_pk$ 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 $\mathbb{Z}_pk$

#### References

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#### Cited by

1. On quasi-cyclic codes over $${\mathbb{Z}_q}$$ vol.20, pp.5-6, 2009, https://doi.org/10.1007/s00200-009-0110-8
2. IDEALS OF Zpn[X]/(Xl-1) vol.26, pp.3, 2011, https://doi.org/10.4134/CKMS.2011.26.3.427
3. CYCLIC CODES OF LENGTH 2nOVER ℤ4 vol.28, pp.1, 2013, https://doi.org/10.4134/CKMS.2013.28.1.039