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The homogeneous distance of \((1+u^2)\)-constacyclic codes over \(\mathbb {F}_2[u]/\langle u^3\rangle \) and its applications

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Abstract

For any \((1+u^2)\)-constacyclic code C with arbitrary length N over \(\mathbb {F}_2[u]/\langle u^3\rangle \), we construct a new map \(\Phi \) from \((\mathbb {F}_2[u]/\langle u^3\rangle )^{N}\) to \((\mathbb {F}_2[v]/\langle v^2\rangle )^{2N}\), and derive \(\Phi (C)\) by applying the map to each codeword of C. We find that C and its image \(\Phi (C)\) have the same homogeneous distance, and the cyclic shift of any codeword of \(\Phi (C)\) is still contained in \(\Phi (C)\) over \(\mathbb {F}_2[v]/\langle v^2\rangle \), but \(\Phi (C)\) is not a cyclic code in some cases. These properties help us to determine the lower bound and the upper bound for the homogeneous distance of C by discussing the same problem of \(\Phi (C)\). With these bounds, exact homogeneous distances of some \((1+u^2)\)-constacyclic codes are given. On the other hand, we consider the application of homogeneous distances in computing Hamming distances of a class of binary linear quasi-cyclic codes, and derive some optimal binary linear quasi-cyclic codes.

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Authors and Affiliations

  1. College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350117, China

    Jian Ding & Changlu Lin

  2. Department of Liberal Education, Anhui Xinhua University, Hefei, 230088, China

    Hongju Li

  3. Guangxi Key Laboratory of Trusted Software, Guilin University of Electronic Technology, Guilin, 541004, China

    Changlu Lin

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  1. Jian Ding
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  2. Hongju Li
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  3. Changlu Lin
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Correspondence to Changlu Lin.

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The research is supported by National Natural Science Foundation of China (U1705264 and 61572132), Natural Science Foundation of Fujian Province (2019J01275), Guangxi Key Laboratory of Trusted Software (KX202039), and University Natural Science Research Project of Anhui Province (KJ20180584, KJ2020A0779)

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Ding, J., Li, H. & Lin, C. The homogeneous distance of \((1+u^2)\)-constacyclic codes over \(\mathbb {F}_2[u]/\langle u^3\rangle \) and its applications. J. Appl. Math. Comput. 68, 83–100 (2022). https://doi.org/10.1007/s12190-020-01475-9

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