How Long Can Optimal Locally Repairable Codes Be?

A locally repairable code (LRC) with locality <inline-formula> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula> allows for the recovery of any erased codeword symbol using only <inline-formula> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula> 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 <italic>optimal</italic>. 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 <inline-formula> <tex-math notation="LaTeX">$ {d} { \geqslant } {5}$ </tex-math></inline-formula>, the code length <inline-formula> <tex-math notation="LaTeX">$ {n}$ </tex-math></inline-formula> of an optimal LRC over an alphabet of size <inline-formula> <tex-math notation="LaTeX">$ {q}$ </tex-math></inline-formula> must be at most roughly <inline-formula> <tex-math notation="LaTeX">$ {O}{(}{{dq}}^{{3}}{)}$ </tex-math></inline-formula>. For the case <inline-formula> <tex-math notation="LaTeX">$ {d} = {5}$ </tex-math></inline-formula>, our upper bound is <inline-formula> <tex-math notation="LaTeX">$ {O}( {q}^{{2}})$ </tex-math></inline-formula>. We complement these bounds by showing the existence of optimal LRCs of length <inline-formula> <tex-math notation="LaTeX">$ {\Omega }_{ {d}, {r}}( {q}^{ {1+1}/\lfloor ( {d}- {3})/ {2}\rfloor }{)}$ </tex-math></inline-formula> when <inline-formula> <tex-math notation="LaTeX">$ {d} \leqslant {r}+ {2}$ </tex-math></inline-formula>. These bounds match when <inline-formula> <tex-math notation="LaTeX">$ {d} = {5}$ </tex-math></inline-formula>, thus pinning down <inline-formula> <tex-math notation="LaTeX">$ {n} = {\Theta }( {q}^{{2}})$ </tex-math></inline-formula> as the asymptotically largest length of an optimal LRC for this case.

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