Some new constructions of optimal and almost optimal locally repairable codes

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 $m_0\ge 2$ and a prime power q, we provide a method to construct optimal and almost optimal $F_q$-additive LRCs over $F_{q^{m_0}}$ with locality r that relies on the existence of certain special polynomials over $F_q$, which we shall refer to as $(r,m_0)$-good polynomials over $F_q$, (note that $(r,m_0)$-good polynomials over $F_q$ coincide with r-good polynomials over $F_q$ when $m_0=1$). We also derive sufficient conditions under which $F_q$-additive LRCs over $F_{q^{m_0}}$ constructed using the aforementioned method are optimal. We further provide four general methods to construct $(r,m_0)$-good polynomials over $F_q$, which give rise to several classes of optimal and almost optimal LRCs over $F_{q^{m_0}}$ with locality r. To illustrate these results, we list several optimal LRCs over $F_{q^{m_0}}$ with new parameters. Finally, we consider the case $m_0=1$ and obtain some new r-good polynomials over $F_q$, which give rise to a construction of optimal linear LRCs over $F_q$ 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.