Constructions of Optimal Cyclic $({r},{\delta })$ Locally Repairable Codes

A code is said to be an <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-local locally repairable code (LRC) if each of its coordinates can be repaired by accessing at most <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> other coordinates. When some of the <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> coordinates are also erased, the <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-local LRC cannot accomplish the local repair, which leads to the concept of <inline-formula> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula>-locality. A <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary <inline-formula> <tex-math notation="LaTeX">$[n, k]$ </tex-math></inline-formula> linear code <inline-formula> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> is said to have <inline-formula> <tex-math notation="LaTeX">$(r, \delta)$ </tex-math></inline-formula>-locality (<inline-formula> <tex-math notation="LaTeX">$\delta \ge 2$ </tex-math></inline-formula>) if for each coordinate <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>, there exists a punctured subcode of <inline-formula> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> with support containing <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>, whose length is at most <inline-formula> <tex-math notation="LaTeX">$r + \delta - 1$ </tex-math></inline-formula>, and whose minimum distance is at least <inline-formula> <tex-math notation="LaTeX">$\delta$ </tex-math></inline-formula>. The <inline-formula> <tex-math notation="LaTeX">$(r, \delta)$ </tex-math></inline-formula>-LRC can tolerate <inline-formula> <tex-math notation="LaTeX">$\delta -1$ </tex-math></inline-formula> erasures in every local code (i.e., punctured subcode), which degenerates to an <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-local LRC when <inline-formula> <tex-math notation="LaTeX">$\delta =2$ </tex-math></inline-formula>. A <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary <inline-formula> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula> LRC is called optimal if it meets the singleton-like bound for <inline-formula> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula>-LRCs. A class of optimal <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary cyclic <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-local LRCs with lengths <inline-formula> <tex-math notation="LaTeX">$n\mid q-1$ </tex-math></inline-formula> were constructed by Tamo, Barg, Goparaju, and Calderbank based on the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary Reed-Solomon codes. In this paper, we construct a class of optimal <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary cyclic <inline-formula> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula>-LRCs (<inline-formula> <tex-math notation="LaTeX">$\delta \ge 2$ </tex-math></inline-formula>) with length <inline-formula> <tex-math notation="LaTeX">$n\mid q-1$ </tex-math></inline-formula>, which generalizes the results of Tamo <italic>et al.</italic> Moreover, we construct a new class of optimal <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary cyclic <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-local LRCs with lengths <inline-formula> <tex-math notation="LaTeX">$n\mid q+1$ </tex-math></inline-formula> and a new class of optimal <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary cyclic <inline-formula> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula>-LRCs (<inline-formula> <tex-math notation="LaTeX">$\delta \ge 2$ </tex-math></inline-formula>) with lengths <inline-formula> <tex-math notation="LaTeX">$n\mid q+1$ </tex-math></inline-formula>. The constructed optimal LRCs with length <inline-formula> <tex-math notation="LaTeX">$n=q+1$ </tex-math></inline-formula> have the best-known length for a given finite field with size <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> when the minimum distance is larger than 4.