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 d5 is upper bounded by O(dq3). Thus, it is a challenging problem to construct q-ary locally recoverable codes with distance d5 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=Ωr(q2) for d=5 and 6, where Ωr means that the implicit constant depends on locality r. In the present 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 (i) our construction is simpler and more explicit; and (ii) the length of our codes is greater than previously known.