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Abstract
It is significant to investigate locally repairable codes (LRCs) since they have abundant applications to distributed storage systems. A locally repairable code is called optimal if it achieves the Singleton-like bound. In [27], I. Tamo et al. presented sufficient and necessary conditions for optimal LRCs. However, it is difficult to check these conditions for many parameters. In this work, we show that construction of optimal LRCs is equivalent to construction of a class of matrices with some conditions. With the help of these conditions, we completely determine all the optimal LRCs for some small parameters. In addition, by shortening technique for the matrices with these conditions, we can easily obtain many new optimal LRCs remaining the minimum distance.
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