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    Constacyclic locally recoverable codes from their duals
    (Springer Heidelberg, 2024) Zengin, Rabia; Koroglu, Mehmet Emin
    A code has locality r if a symbol in any coordinate of a codeword in the code can be recovered by accessing the value of at most r other coordinates. Such codes are called locally recoverable codes (LRCs for short). Since LRCs can recover a failed node by accessing the minimum number of the surviving nodes, these codes are used in distributed storage systems such as Microsoft Azure. In this paper, constacyclic LRCs are obtained from their parity-check polynomials. Constacyclic codes with locality r <= 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\le 2$$\end{document} and dimension 2 are obtained and a sufficient and necessary condition for these codes to have locality 1 is given. Then, the construction is generalized. Distance optimal constacyclic LRCs with distance 2 are obtained. Also, constacyclic codes with locality 1 are constructed. They may be so useful in practice thanks to their minimum locality. Constacyclic codes whose locality is equal to their dimension are given. Furthermore, constacyclic LRCs are obtained from cyclotomic cosets.
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    Period-3 Orbits of Sequential Dynamical Systems and Their Relationship to Error-Correcting Codes over Finite Fields
    (Springer Japan Kk, 2026) Ulutas, Tugce; Koroglu, Mehmet Emin
    A sequential dynamical system consists of the following data; a finite graph Y with vertex set v1,& mldr;,vn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1,\ldots ,v_n$$\end{document}, a state set, local update functions, and an update ordering sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}. We study period-3 orbits of SDSs on the complete graph Kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_n$$\end{document} with identical local functions. We prove that the maximum number theta n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{n+1}$$\end{document} of 3-cycles in the phase space equals A3(n,4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_3(n,4)$$\end{document}, the largest size of a ternary code of length n with minimum Hamming distance at least 4. Our approach reduces the problem to a clique number computation in an explicit graph and yields a direct correspondence with optimal ternary (n, 4)-codes. We also give field-agnostic necessary conditions for prime period-p orbits and discuss extensions over Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_p$$\end{document}.