Some new constructions of optimal and almost optimal locally repairable codes

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Additive codes over finite fields are natural extensions of linear codes and are useful in constructing quantum error-correcting codes. In this paper, we first study the locality properties of additive MDS codes over finite fields whose dual codes are also MDS. We further provide a method to construct optimal and almost optimal LRCs with new parameters belonging to the family of additive codes, which are not MDS. More precisely, for an integer m0≥2 and a prime power q, we provide a method to construct optimal and almost optimal Fq-additive LRCs over Fqm0 with locality r that relies on the existence of certain special polynomials over Fq, which we shall refer to as (r,m0)-good polynomials over Fq, (note that (r,m0)-good polynomials over Fq coincide with r-good polynomials over Fq when m0=1). We also derive sufficient conditions under which Fq-additive LRCs over Fqm0 constructed using the aforementioned method are optimal. We further provide four general methods to construct (r,m0)-good polynomials over Fq, which give rise to several classes of optimal and almost optimal LRCs over Fqm0 with locality r. To illustrate these results, we list several optimal LRCs over Fqm0 with new parameters. Finally, we consider the case m0=1 and obtain some new r-good polynomials over Fq, which give rise to a construction of optimal linear LRCs over Fq with locality r. We illustrate this result by listing several optimal linear LRCs over smaller finite fields compared to the previously known LRCs with the same parameters.