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Erschienen in:
19.01.2021 | Original Paper
On the symbol-pair distance of some classes of repeated-root constacyclic codes over Galois ring
Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 1/2023
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Abstract
Let \(\gamma = 4z-1\) be an unit of Type \((*^{-})\) of the Galois ring \({{\,\mathrm{GR}\,}}(2^a, m)\). The \(\gamma\)-constacyclic codes of length \(2^s\) over the Galois ring \({{\,\mathrm{GR}\,}}(2^a, m)\) are precisely the ideals \(\langle (x +1)^i \rangle\), \(0 \le i \le 2^sa\) of the chain ring \(\mathfrak {R}(a,m, \gamma ) = \dfrac{{{\,\mathrm{GR}\,}}(2^a,m)[x]}{\langle {x^{2^s}} - \gamma \rangle }\). This structure is used to determine the symbol pair distance of \(\gamma\)-constacyclic codes of length \(2^s\) over \({{\,\mathrm{GR}\,}}(2^a, m)\). The exact symbol-pair distances for all such \(\gamma\)-constacyclic codes of length \(2^s\) over \({{\,\mathrm{GR}\,}}(2^a, m)\) are obtained. Also, we provide the MDS symbol-pair codes of length \(2^s\) over \({{\,\mathrm{GR}\,}}(2^a, m)\) and some examples are computed.