Explicit Constructions of Optimal-Access MSCR Codes for All Parameters

The MSCR (minimum storage cooperative regenerating) code is an important variation of regenerating codes for repairing multiple node failures in a cooperative way and retaining the minimum storage. However, explicit constructions of MSCR codes for all parameters were not developed untill Ye&Barg’s recent work. Here we present another explicit construction of MSCR codes for all parameters. Specifically, we design an <inline-formula> <tex-math notation="LaTeX">$({n},{k})$ </tex-math></inline-formula> MDS code that can cooperatively repair any <inline-formula> <tex-math notation="LaTeX">${h}\,\,(2 \leq {h} \leq {n}-{k})$ </tex-math></inline-formula> erasures from any <inline-formula> <tex-math notation="LaTeX">${d}\,\,({k} \leq {d} \leq {n}-{h})$ </tex-math></inline-formula> helper nodes. The superiority of our code to Ye&Barg’s is three fold: (1) Our code additionally has the optimal access property, i.e., the amount of data accessed at each helper node meets a lower bound on this quantity; (2) The sub-packetization level is reduced by a factor of <inline-formula> <tex-math notation="LaTeX">$(d-k)^{(h-1)\binom {n}{h}}$ </tex-math></inline-formula>; (3) Our code is built over a smaller field <inline-formula> <tex-math notation="LaTeX">${F}$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$|{F}|\geq {n}+{d}-{k}$ </tex-math></inline-formula>, compared to Ye&Barg’s requirement <inline-formula> <tex-math notation="LaTeX">$|{F}|\geq ({d}-{k}+1){n}$ </tex-math></inline-formula>.