Explicit Constructions of Optimal-Access MDS Codes With Nearly Optimal Sub-Packetization

An <inline-formula> <tex-math notation="LaTeX">$(n,k,l)$ </tex-math></inline-formula> maximum distance separable (MDS) array code of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, dimension <inline-formula> <tex-math notation="LaTeX">$k=n-r$ </tex-math></inline-formula>, and sub-packetization <inline-formula> <tex-math notation="LaTeX">$l$ </tex-math></inline-formula> is formed of <inline-formula> <tex-math notation="LaTeX">$l\times n$ </tex-math></inline-formula> matrices over a finite field <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>, with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword. Repair of a failed node (recovery of one erased column) can be performed by accessing a set of <inline-formula> <tex-math notation="LaTeX">$d\le n-1$ </tex-math></inline-formula> surviving (helper) nodes. The code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity. For optimal-access MDS codes with <inline-formula> <tex-math notation="LaTeX">$d=n-1$ </tex-math></inline-formula>, the sub-packetization <inline-formula> <tex-math notation="LaTeX">$l$ </tex-math></inline-formula> satisfies the bound <inline-formula> <tex-math notation="LaTeX">$l\ge r^{(k-1)/r}$ </tex-math></inline-formula>. In our previous work (IEEE Trans. Inf. Theory, vol. 63, no. 4, 2017), for any <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>, we presented an explicit construction of optimal-access MDS codes with sub-packetization <inline-formula> <tex-math notation="LaTeX">$l=r^{n-1}$ </tex-math></inline-formula>. In this paper, we take up the question of reducing the sub-packetization value <inline-formula> <tex-math notation="LaTeX">$l$ </tex-math></inline-formula> to make it to approach the lower bound. We construct an explicit family of optimal-access codes with <inline-formula> <tex-math notation="LaTeX">$l=r^{\lceil n/r\rceil }$ </tex-math></inline-formula>, which differs from the optimal value by at most a factor of <inline-formula> <tex-math notation="LaTeX">$r^{2}$ </tex-math></inline-formula>. These codes can be constructed over any finite field <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> as long as <inline-formula> <tex-math notation="LaTeX">$|F|\ge r\lceil n/r\rceil $ </tex-math></inline-formula>, and afford low-complexity encoding and decoding procedures. We also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints (LRC codes), which we call <italic>group repair with optimal access</italic>. In this variation, we assume that the set of <inline-formula> <tex-math notation="LaTeX">$n=sm$ </tex-math></inline-formula> nodes is partitioned into <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> repair groups of size <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>, and require that the amount of accessed data for repair is the smallest possible whenever the <inline-formula> <tex-math notation="LaTeX">$d=s+k-1$ </tex-math></inline-formula> helper nodes include all the other <inline-formula> <tex-math notation="LaTeX">$s-1$ </tex-math></inline-formula> nodes from the same group as the failed node. For this problem, we construct a family of codes with the group optimal access property. These codes can be constructed over any field <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> of size <inline-formula> <tex-math notation="LaTeX">$|F|\ge n$ </tex-math></inline-formula>, and also afford low-complexity encoding and decoding procedures.