Explicit Construction of Optimal Locally Recoverable Codes of Distance 5 and 6 via Binary Constant Weight Codes

In a paper by Guruswami et al., it was shown that the length n of a q-ary linear locally recoverable code with distance d ≥ 5 is upper bounded by O(dq
³
). Thus, it is a challenging problem to construct q-ary locally recoverable codes with distance d ≥ 5 and length approaching the upper bound. The same paper also gave an algorithmic construction of q-ary locally recoverable codes with locality r and length n = Ω
_r
(q
²
) for d = 5 and 6, where Ωr means that the implicit constant depends on locality r. In this paper, we present an explicit construction of q-ary locally recoverable codes of distance d = 5 and 6 via binary constant weight codes. It turns out that 1) our construction is simpler and more explicit and 2) the length of our codes is greater than previously known.