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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]

In this paper, we use the CSS and Steane's constructions to establish quantum error-correcting codes (briefly, QEC codes) from cyclic codes of length 3 p s over F p m . We obtain several new classes of QEC codes in the sense that their parameters are different from all the previous constructions. Among them, we identify all quantum MDS (briefly, qMDS) codes, i.e., optimal quantum codes with respect to the quantum Singleton bound. In addition, we construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length 3 p s over F p m . Furthermore, we give many new QSCs to enrich the variety of available QSCs. A lot of them are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes. [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 establish all quantum error-correcting codes (briefly, QEC codes) from cyclic and negacyclic codes of length 4 p s over F p m using CSS and Steane's constructions. Some new QEC codes are given in the sense that their parameters are different from all the previous constructions. In addition, if we fix the length and the Hamming distance of a QEC code, some QEC codes are better than all QEC codes in the sense that their dimensions are larger than the dimensions of all the previous QEC codes. We also investigate quantum maximum distance separable (briefly, qMDS) codes from cyclic and negacyclic codes of length 4 p s over finite fields using the CSS and Hermitian constructions. We provide all qMDS codes constructed from dual codes of cyclic and negacyclic codes of length 4 p s over finite fields using the Hermitian construction. We also construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length 4 p s over F p m . 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 non-primitive narrow-sense BCH codes. [ABSTRACT FROM AUTHOR]

Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson–Gorenstein–Zierler's algorithm, is designed for them. [ABSTRACT FROM AUTHOR]

Let 𝔽 p m be a finite field with pm elements for an odd prime p and a positive integer m. In this paper, we aim to construct non-binary quantum codes from repeated root cyclic and negacyclic codes of length 4ps over 𝔽 p m . To achieve this, first we discuss the structure of all maximum distance separable (MDS) cyclic and negacyclic codes of length 4ps over 𝔽 p m and their Euclidean and Hermitian duals. Further, we establish a necessary and sufficient condition for these codes to contain their duals. Finally, many new and quantum MDS codes are obtained with the help of CSS and Hermitian constructions from dual containing cyclic and negacyclic codes. [ABSTRACT FROM AUTHOR]

Decoding algorithms for the correction of errors for cyclic codes over quaternion integers of quaternion Mannheim weight one up to two coordinates are discussed by Özen and Güzeltepe (Eur J Pure Appl Math 3(4):670–677, 2010 ). Though, Neto et al. (IEEE Trans Inf Theory 47(4):1514–1527, 2001 ) proposed decoding algorithms for the correction of errors of arbitrary Mannheim weight. In this study, we followed the procedures used by Neto et al. and suggest a decoding algorithm for an $$n$$ n length cyclic code over quaternion integers to correct errors of quaternion Mannheim weight two up to two coordinates. Furthermore, we establish that; over quaternion integers, for a given $$n$$ n length cyclic code there exist a cyclic code of length $$2n-1$$ 2 n - 1 . The decoding algorithms for the cyclic code of length $$2n-1$$ 2 n - 1 are given, which correct errors of quaternion Mannheim weight one and two. In addition, we show that the cyclic code of length $$2n-1$$ 2 n - 1 is maximum-distance separable (MDS) with respect to Hamming distance. [ABSTRACT FROM AUTHOR]