MSR codes with linear field size and smallest sub-packetization for any number of helper nodes

An $(n,k,\ell)$ array code has $k$ information coordinates and $r = n-k$ parity coordinates, where each coordinate is a vector in $\mathbb{F}_q^{\ell}$ for some field $\mathbb{F}_q$. An $(n,k,\ell)$ 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) array 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 $n-1$ 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 $\ell$ 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 $\ell\geq s^{\left\lceil n/s \right\rceil}$, 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, till now, reduction to linear field size is only available for $d\in\{k+1,k+2,k+3\}$ and $d=n-1$. In this paper, we construct optimal-access MSR codes with linear field size and smallest sub-packetization $\ell = s^{\left\lceil n/s \right\rceil}$ for all $d$ between $k+1$ and $n-1$. We also construct another class of MSR codes that are not optimal-access but have even smaller sub-packetization $s^{\left\lceil n/(s+1)\right\rceil }$. The second class also has linear field size and works for all admissible values of $d$.