MSR Codes With Linear Field Size and Smallest Sub-Packetization for Any Number of Helper Nodes

An <inline-formula> <tex-math notation="LaTeX">$(n, k, \ell)$ </tex-math></inline-formula> array code has k information coordinates and <inline-formula> <tex-math notation="LaTeX">$r = n - k$ </tex-math></inline-formula> parity coordinates, where each coordinate is a vector in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}^{\ell }$ </tex-math></inline-formula> for some finite field <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>. An <inline-formula> <tex-math notation="LaTeX">$(n, k, \ell)$ </tex-math></inline-formula> MDS array code has the additional property that any k out of n coordinates suffice to recover the whole codeword. Dimakis et al. considered the problem of repairing the erasure of a single coordinate and proved a lower bound on the amount of data transmission that is needed for the repair. A minimum storage regenerating (MSR) code with repair degree d is an MDS array code that achieves this lower bound for the repair of any single erased coordinate from any d out of <inline-formula> <tex-math notation="LaTeX">$n-1$ </tex-math></inline-formula> remaining coordinates. An MSR code has the optimal access property if the amount of accessed data is the same as the amount of transmitted data in the repair procedure. The sub-packetization <inline-formula> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> and the field size q are of paramount importance in MSR code constructions. For optimal-access MSR codes, Balaji et al. proved that <inline-formula> <tex-math notation="LaTeX">$\ell \geq s^{\left \lceil {{ n/s }}\right \rceil }$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$s = d-k+1$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$d\in \{k+1,k+2,k+3\}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$d=n-1$ </tex-math></inline-formula>. In this paper, we construct the first class of explicit optimal-access MSR codes with the smallest sub-packetization <inline-formula> <tex-math notation="LaTeX">$\ell = s^{\left \lceil {{ n/s }}\right \rceil }$ </tex-math></inline-formula> for all d between <inline-formula> <tex-math notation="LaTeX">$k+1$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n-1$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$s^{\left \lceil {{ n/(s+1)}}\right \rceil }$ </tex-math></inline-formula> for all admissible values of d, making significant progress on another open problem in the survey. Previously, MSR codes with <inline-formula> <tex-math notation="LaTeX">$\ell =s^{\left \lceil {{ n/(s+1)}}\right \rceil }$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$q=O(n)$ </tex-math></inline-formula> were only known for <inline-formula> <tex-math notation="LaTeX">$d=k+1$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$d=n-1$ </tex-math></inline-formula>. The key insight that enables a linear field size in our construction is to reduce <inline-formula> <tex-math notation="LaTeX">$\binom {n}{r}$ </tex-math></inline-formula> global constraints of non-vanishing determinants to <inline-formula> <tex-math notation="LaTeX">$O_{s}(n)$ </tex-math></inline-formula> local ones, which is achieved by carefully designing the parity check matrices.