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.
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Acknowledgment
This work is a part of Ph.D. thesis of Manoj Kumar Singh. The authors are grateful to the anonymous reviewers for helpful comments to improve the manuscript. This work was supported in part by the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Thailand.
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Dinh, H.Q., Kumar, N., Singh, A.K. et al. On the symbol-pair distance of some classes of repeated-root constacyclic codes over Galois ring. AAECC 34, 111–128 (2023). https://doi.org/10.1007/s00200-020-00472-6
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DOI: https://doi.org/10.1007/s00200-020-00472-6