Optimal Locally Repairable Codes for Parallel Reading

Locally repairable codes (LRCs) have important applications in distributed storage systems. In this paper, we study q-ary [n, k, d] LRCs with (r, t, δ)-information-locality, where each of the i-th (1 ≤ i ≤ k) information symbol is contained in t punctured subcodes with length ≤ r + δ - 1, minimum distance δ, and the i-th information symbol is the unique common code symbol of these t subcodes, furthermore, each subcode contains exactly δ - 1 parity symbols. Firstly, an upper bound on the minimum distance of such q-ary LRCs with (r, t, δ)-information-locality is given. Then, we propose a general construction framework of q-ary optimal LRCs with (r, t, δ)-information-locality and minimum distanced = t(δ-1)+1, where the required field size is just q ≥ r+δ-2. The proposed optimal LRCs can always repair a failed information node locally in case of at most tδ - 1 node failures. Moreover, multiple repair subcodes can support parallel readings of data, thus make the proposed codes attractive for distributed storage systems with hot data.