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On modular cyclic codes. (English) Zbl 1130.94333
Summary: We study cyclic codes of arbitrary length $$N$$ over the ring of integers modulo $$M$$. We first reduce this to the study of cyclic codes of length $$N=p^kn$$ ($$n$$ prime to $$p$$) over the ring $$\mathbb{Z}_{p^e}$$ for prime divisors $$p$$ of $$N$$. We then use the discrete Fourier transform to obtain an isomorphism $$\gamma$$ between $$\mathbb{Z}_{p^e}[X]/(X^N-1)$$ and a direct sum $$\bigoplus_{i\in I}{\mathcal S}_i$$ of certain local rings which are ambient spaces for codes of length $$p^k$$ over certain Galois rings, where $$I$$ is the complete set of representatives of $$p$$-cyclotomic cosets modulo $$n$$. Via this isomorphism we may obtain all codes over $$\mathbb{Z}_{p^e}$$ from the ideals of $${\mathcal S}_i$$. The inverse isomorphism of $$\gamma$$ is explicitly determined, so that the polynomial representations of the corresponding ideals can be calculated. The general notion of higher torsion codes is defined and the ideals of $$\mathcal S_i$$ are classified in terms of the sequence of their torsion codes.

##### MSC:
 94B15 Cyclic codes 94B05 Linear codes, general 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 14H20 Singularities of curves, local rings
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