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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 &#x2265; 5 is upper bounded by O(dq3). Thus, it is a challenging problem to construct q-ary locally recoverable codes with distance d &#x2265; 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 = &#x03A9;r(q2) for d = 5 and 6, where &#x03A9;r 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.

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