New Construction of Locally q-ary Sequential Recoverable Codes: Parity-check Matrix Approach
Akram Baghban, Mehdi Ghiyasvand
TL;DR
This work addresses recovering multiple failures in distributed storage by introducing sequential locally recoverable codes (SLRCs) with information $(r,t_i,\delta)$-sequential-locality. It develops a parity-check-matrix construction that links parallel and sequential recovery, enabling sequential repair of up to $t \ge \delta t_i+1$ erasures over any $q$-ary field and at most $r$ symbols per recovery. The key contributions include: (i) defining information $(r,t_i,\delta)$-SLRC, (ii) Algorithm 1 for constructing $H$ from a local $[r+\delta-1,r,\delta]$-MDS code and a resolvable-design incidence matrix, and (iii) a rate expression $k/n=(1+\lceil t/r\rceil+\lceil 1/r^2\rceil(\delta-1))^{-1}$ along with comparisons showing broad applicability beyond binary alphabets. The results advance practical SLRC design for larger $t$ and arbitrary $q$, with potential impact on the efficiency and reliability of distributed storage systems.
Abstract
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 $(r, t_i, δ)$-sequential-locality where each of the $i$-th information symbols is contained in $t_i$ punctured subcodes with length $(r+δ-1)$ and minimum distance $δ$. We prove that such codes are $(r, t)_q$-SLRC ($t \geq δt_i+1$), which implies that they permit sequential recovery for up to $t$ erasures each one by $r$ other code symbols.