A Construction of Optimal Quasi-cyclic Locally Recoverable Codes using Constituent Codes
| Autor: | Bastos, Gustavo Terra, Alvarez, Angelynn, Flores, Zachary, Salerno, Adriana |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | A locally recoverable code of locality $r$ over $\mathbb{F}_{q}$ is a code where every coordinate of a codeword can be recovered using the values of at most $r$ other coordinates of that codeword. Locally recoverable codes are efficient at restoring corrupted messages and data which make them highly applicable to distributed storage systems. Quasi-cyclic codes of length $n=m\ell$ and index $\ell$ are linear codes that are invariant under cyclic shifts by $\ell$ places. %Quasi-cyclic codes are generalizations of cyclic codes and are isomorphic to $\mathbb{F}_{q} [x]/ \langle x^m-1 \rangle$-submodules of $\mathbb{F}_{q^\ell} [x] / \langle x^m-1 \rangle$. In this paper, we decompose quasi-cyclic locally recoverable codes into a sum of constituent codes where each constituent code is a linear code over a field extension of $\mathbb{F}_q$. Using these constituent codes with set parameters, we propose conditions which ensure the existence of almost optimal and optimal quasi-cyclic locally recoverable codes with increased dimension and code length. Comment: 12 pages |
| Databáze: | arXiv |
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