MSR Codes with Linear Field Size and Smallest Sub-packetization for Any Number of Helper Nodes
Guodong Li, Ningning Wang, Sihuang Hu, Min Ye · 2024
The sub-packetization$\ell$and the field size$q$are of paramount importance in MSR code constructions. For optimal-access MSR codes, Balaji et al. proved that$\ell\geq s^{\lceil n/s\rceil}$, where$s= d-k+1$. Rawat et al. showed that this lower bound is attainable for all admissible values of$d$when the field size is exponential in$n$. After that, tremendous efforts have been devoted to reducing the field size. However, so far, reduction to a linear field size is only available for$d\in\{k+1, k+2, k+3\}$and$d=n-1$. In this paper, we construct the first class of explicit ontimal-access MSR codes with the smallest sub-packetization$\ell=s^{\lceil n/s\rceil}$for all$d$between$k+1$and$n-1$, resolving an open problem in the survey (Ramkumar et al., Foundations and Trends in Communications and Information Theory: Vol. 19: No. 4). We further propose another class of explicit MSR code constructions (not optimal-access) with an even smaller sub-packetization$s^{\lceil n/(s+1)\rceil}$for all admissible values of$d$, making significant progress on another open problem in the survey. Previously, MSR codes with$\ell= s^{\lceil n/(s+1)\rceil}$and$q=O(n)$were only known for$d=k+1$and$d=n-1$. The key insight that enables a linear field size in our construction is to reduce$\binom{n}{r}$global constraints of non-vanishing determinants to$O_{s}(n)$local ones, which is achieved by carefully designing a group of kernel matrices and then blowing them up to get parity check submatrices.