Matrix-product constructions giving rise to locally recoverable codes are considered, both the classical $r$ and $(r,\delta)$ localities. We study the recovery advantages offered by the constituent codes and also by the defining matrices of the matrix product codes. Finally, we extend these methods to a particular variation of matrix-product codes and quasi-cyclic codes. Singleton-optimal locally recoverable codes and almost Singleton-optimal codes, with length larger than the finite field size, are obtained, some of them with superlinear length.

翻译：本文研究了由矩阵积构造产生的局部修复码，涵盖经典$r$局部性与$(r,\delta)$局部性。我们分析了组成码以及矩阵积码定义矩阵所提供的修复优势。最后，我们将这些方法推广到矩阵积码和准循环码的特定变体中。由此得到了长度大于有限域大小的Singleton最优局部修复码和几乎Singleton最优码，其中部分码具有超线性长度。