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

A locally recoverable code of locality r over Fq 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 and index are linear codes that are invariant under cyclic shifts by places. %Quasi-cyclic codes are generalizations of cyclic codes and are isomorphic to Fq [x]/ xᵐ-1-submodules of Fq^ [x] / xᵐ-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 Fq. 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.