Explicit MSR Codes with Optimal Access, Optimal Sub-Packetization and Small Field Size for $d = k+1, k+2, k+3$

This paper presents the construction of an explicit, optimal-access, high-rate MSR code for any $(n,k,d=k+1,k+2,k+3)$ parameters over the finite field $fQ$ having sub-packetization $alpha = q^{lceilfrac{n}{q}rceil}$, where $q=d-k+1$ and $Q = O(n)$. The sub-packetization of the current construction meets the lower bound proven in a recent work by Balaji et al. in cite{BalKum}. To our understanding the codes presented in this paper are the first explicit constructions of MSR codes with $d<(n-1)$ having optimal sub-packetization, optimal access and small field size.