Improved Maximally Recoverable LRCs using Skew Polynomials

An $(n,r,h,a,q)$-Local Reconstruction Code (LRC) is a linear code over ${\mathbb{F}}_q$ of length $n$, whose codeword symbols are partitioned into $n\mathrm{/}r$ local groups each of size $r$. Each local group satisfies $a$' local parity checks to recover from $a$’ erasures in that local group and there are further $h$ global parity checks to provide fault tolerance from more global erasure patterns. Such an LRC is Maximally Recoverable (MR), if it offers the best blend of locality and global erasure resilience – namely it can correct all erasure patterns whose recovery is information-theoretically feasible given the locality structure (these are precisely patterns with up to `$a$’ erasures in each local group and an additional $h$ erasures anywhere in the codeword). Random constructions can easily show the existence of MR LRCs over very large fields, but a major algebraic challenge is to construct MR LRCs, or even show their existence, over smaller fields, as well as understand inherent lower bounds on their field size. We give an explicit construction of $(n,r,h,a,q)$-MR LRCs with field size $q$ bounded by ${\left(O(\max \{r,n\mathrm{/}r\})\right)}^{\min \{h,r-a\}}$. This improves upon known constructions in many relevant parameter ranges. Moreover, it matches the lower bound from Gopi et al. (2020) in an interesting range of parameters where $r=\mathrm{\Theta }(\sqrt{n})$, $r-a=\mathrm{\Theta }(\sqrt{n})$ and $h$ is a fixed constant with $h\le a+2$, achieving the optimal field size of ${\mathrm{\Theta }}_h(n^{h\mathrm{/}2})\mathrm{.}$ Our construction is based on the theory of skew polynomials. We believe skew polynomials should have further applications in coding and complexity theory; as a small illustration we show how to capture algebraic results underlying list decoding folded Reed-Solomon and multiplicity codes in a unified way within this theory.