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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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