How Long Can Optimal Locally Repairable Codes Be?

A locally repairable code (LRC) with locality ${r}$ allows for the recovery of any erased codeword symbol using only ${r}$ other codeword symbols. A Singleton-type bound dictates the best possible tradeoff between the dimension and distance of LRCs—an LRC attaining this tradeoff is deemed optimal . Such optimal LRCs have been constructed over alphabets growing linearly in the block length. Unlike the classical Singleton bound, however, it was not known if such a linear growth in the alphabet size is necessary or, for that matter, even if the alphabet needs to grow at all with the block length. Indeed, for small code distances 3 and 4, arbitrarily long optimal LRCs were known over fixed alphabets. Here, we prove that for distances $ {d} { \geqslant } {5}$ , the code length $ {n}$ of an optimal LRC over an alphabet of size $ {q}$ must be at most roughly $ {O}{(}{{dq}}^{{3}}{)}$ . For the case $ {d} = {5}$ , our upper bound is $ {O}( {q}^{{2}})$ . We complement these bounds by showing the existence of optimal LRCs of length $ {\Omega }_{ {d}, {r}}( {q}^{ {1+1}/\lfloor ( {d}- {3})/ {2}\rfloor }{)}$ when $ {d} \leqslant {r}+ {2}$ . These bounds match when $ {d} = {5}$ , thus pinning down $ {n} = {\Theta }( {q}^{{2}})$ as the asymptotically largest length of an optimal LRC for this case.