Singleton-optimal (r,δ)-LRCs via some good polynomials of special forms
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RS-like locally repairable codes (LRCs) based on polynomial evaluation were first introduced by Tamo and Barg in 2014. The constructions rely on the so-called good polynomials that is constant on each of some sets, which determine the upper bound of the code length. In this paper, based on the constructions of RS-like LRCs, we propose new constructions of (r,δ)-LRCs via some good polynomials of special forms, whose code length can be slightly larger than the RS-like LRCs. These LRCs are all distance-optimal when the other parameters are fixed and attain the Singleton-type bound for a certain class of dimension k. It is worth noting that the Singleton-optimal (r,δ)-LRCs with length n=q-1+δ can be constructed when r≥ 2 and (r+δ-1)| (q-1).
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