MSR codes with linear field size and smallest sub-packetization for any number of helper nodes
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The sub-packetization ℓ and the field size q are of paramount importance in the MSR array code constructions. For optimal-access MSR codes, Balaji et al. proved that ℓ≥ s^⌈ n/s ⌉, 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, till now, reduction to linear field size is only available for d∈{k+1,k+2,k+3} and d=n-1. In this paper, we construct the first class of explicit optimal-access MSR codes with the smallest sub-packetization ℓ = s^⌈ n/s ⌉ 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 even smaller sub-packetization s^⌈ n/(s+1)⌉ for all admissible values of d, making significant progress on another open problem in the survey. Previously, MSR codes with ℓ=s^⌈ n/(s+1)⌉ 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 nr global constraints of non-vanishing determinants to O_s(n) local ones, which is achieved by carefully designing the parity check matrices.
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