Matrix-product structure of constacyclic codes over finite chain rings \mathbb {F}_{p^m}[u]/\langle u^e\rangle (original) (raw)

Abstract

Let m, e be positive integers, p a prime number, \(\mathbb {F}_{p^m}\) be a finite field of \(p^m\) elements and \(R=\mathbb {F}_{p^m}[u]/\langle u^e\rangle \) which is a finite chain ring. For any \(\omega \in R^\times \) and positive integers k, n satisfying \(\mathrm{gcd}(p,n)=1\), we prove that any \((1+\omega u)\)-constacyclic code of length \(p^kn\) over R is monomially equivalent to a matrix-product code of a nested sequence of \(p^k\) cyclic codes with length n over R and a \(p^k\times p^k\) matrix \(A_{p^k}\) over \(\mathbb {F}_p\). Using the matrix-product structures, we give an iterative construction of every \((1+\omega u)\)-constacyclic code by \((1+\omega u)\)-constacyclic codes of shorter lengths over R.

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Acknowledgements

Part of this work was done when Yonglin Cao was visiting Chern Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality. This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 11671235, 61571243, 11471255).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255091, Shandong, China
    Yuan Cao & Yonglin Cao
  2. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China
    Fang-Wei Fu

Authors

  1. Yuan Cao
  2. Yonglin Cao
  3. Fang-Wei Fu

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Correspondence toYonglin Cao.

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Cao, Y., Cao, Y. & Fu, FW. Matrix-product structure of constacyclic codes over finite chain rings \(\mathbb {F}_{p^m}[u]/\langle u^e\rangle \).AAECC 29, 455–478 (2018). https://doi.org/10.1007/s00200-018-0352-4

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