New Construction of Locally q-ary Sequential Recoverable Codes: Parity-check Matrix Approach

This paper develops a new family of locally recoverable codes for distributed storage systems, Sequential Locally Recoverable Codes (SLRCs) constructed to handle multiple erasures in a sequential recovery approach. We propose a new connection between parallel and sequential recovery, which leads to a general construction of q-ary linear codes with information (r, tᵢ, δ)-sequential-locality where each of the i-th information symbols is contained in tᵢ punctured subcodes with length (r+δ-1) and minimum distance δ. We prove that such codes are (r, t)q-SLRC (t ≥ δ tᵢ+1), which implies that they permit sequential recovery for up to t erasures each one by r other code symbols.