A Construction of Optimal Quasi-cyclic Locally Recoverable Codes using Constituent Codes

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 $n=m\mathrm{\ell }$ and index $\mathrm{\ell }$ are linear codes that are invariant under cyclic shifts by $\mathrm{\ell }$ places. %Quasi-cyclic codes are generalizations of cyclic codes and are isomorphic to ${\mathbb{F}}_q[x]\mathrm{/}⟨x^m-1⟩$-submodules of ${\mathbb{F}}_{q^{\mathrm{\ell }}}[x]\mathrm{/}⟨x^m-1⟩$. 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.