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For any odd prime p ≠ 5 , the structures of cyclic codes of length 5 p s over F p m are applied to construct quantum error-correcting codes (briefly, QEC codes). Some new QEC codes are provided in the sense that their parameters are different from all the previous constructions. We give all quantum maximum-distance-separable (briefly, qMDS codes) constructed by the CSS construction. We also construct quantum synchronizable codes (briefly, QSCs). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense BCH codes. [ABSTRACT FROM AUTHOR]

Let R be the finite chain ring R = F p m + u F p m (u 2 = 0) , where p is an odd prime number and m is a positive integer. For η ∈ F p m ∗ , the Hamming distances of all η -constacyclic codes of length 2 p s over R had already been studied in Dinh et al. (in AAECC, 2020. https://doi.org/10.1007/s00200-020-00432-0). However, such a study is incomplete. In this paper, we provide corrections to some results that appeared in Dinh et al. (2020) and we completely solve the problem of determination of the Hamming distance of η -constacyclic codes of length 2 p s over R . [ABSTRACT FROM AUTHOR]

Let γ = 4 z - 1 be an unit of Type (∗ -) of the Galois ring GR (2 a , m) . The γ -constacyclic codes of length 2 s over the Galois ring GR (2 a , m) are precisely the ideals ⟨ (x + 1) i ⟩ , 0 ≤ i ≤ 2 s a of the chain ring R (a , m , γ) = GR (2 a , m) [ x ] ⟨ x 2 s - γ ⟩ . This structure is used to determine the symbol pair distance of γ -constacyclic codes of length 2 s over GR (2 a , m) . The exact symbol-pair distances for all such γ -constacyclic codes of length 2 s over GR (2 a , m) are obtained. Also, we provide the MDS symbol-pair codes of length 2 s over GR (2 a , m) and some examples are computed. [ABSTRACT FROM AUTHOR]

Cyclic codes over finite fields have been studied for decades due to their wide applicability in communication systems, consumer electronics, and data storage systems. Let p be an odd prime and let s and m be positive integers. In this paper, we first determine the Hamming distances of all cyclic codes of length 8 over Fq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_q$$\end{document}. Building upon this, we explicitly obtain the Hamming distances of all repeated-root cyclic codes of length 8ps\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$8p^s$$\end{document} over Fq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_q$$\end{document}. As an application, we give all maximum distance separable cyclic codes of length 8ps\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$8p^s$$\end{document}. [ABSTRACT FROM AUTHOR]

In this paper we consider an error-detecting code based on linear quasigroups. Namely, each input block a 0 a 1 ... a n - 1 is extended into a block a 0 a 1 ... a n - 1 d 0 d 1 ... d n - 1 , where the redundant characters d 0 , d 1 , ... , d n - 1 are defined with d i = a i ∗ a i + 1 ∗ a i + 2 , where ∗ is a linear quasigroup operation and the operations in the indexes are modulo n. We give a proof that under some conditions the code is linear. Using this fact, we contribute to the determination of the error-detecting capability of the code. Namely, we determine the Hamming distance of the code and from there we obtain the number of errors that the code will detect for sure when linear quasigroups of order 4 from the best class of quasigroups of order 4 for which the constant term in the linear representation is zero matrix are used for coding. All results in the paper are derived for arbitrary length of the input blocks. With the obtained results we showed that when a small linear quasigroup of order 4 from the best class of quasigroups of order 4 is used for coding, the number of errors that the code surely detects is upper bounded with 4. [ABSTRACT FROM AUTHOR]