Improved Maximally Recoverable LRCs Using Skew Polynomials

An <inline-formula> <tex-math notation="LaTeX">$(n,r,h,a,q)$ </tex-math></inline-formula>-Local Reconstruction Code (LRC) is a linear code over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, whose codeword symbols are partitioned into <inline-formula> <tex-math notation="LaTeX">$n/r$ </tex-math></inline-formula> local groups each of size <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>. Each local group satisfies ‘<inline-formula> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula>’ local parity checks to recover from ‘<inline-formula> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula>’ erasures in that local group and there are further <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> 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 ‘<inline-formula> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula>’ erasures in each local group and an additional <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$(n,r,h,a,q)$ </tex-math></inline-formula>-MR LRCs with field size <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> bounded by <inline-formula> <tex-math notation="LaTeX">$\left ({O\left ({\max \{r,n/r\}}\right)}\right)^{\min \{h,r-a\}}$ </tex-math></inline-formula>. This significantly improves upon known constructions in many practically relevant parameter ranges. Moreover, it matches the lower bound from Gopi et al. (2020) in an interesting range of parameters where <inline-formula> <tex-math notation="LaTeX">$r=\Theta (\sqrt {n})$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$r-a=\Theta (\sqrt {n})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> is a fixed constant with <inline-formula> <tex-math notation="LaTeX">$h \leqslant a+2$ </tex-math></inline-formula>, achieving the optimal field size of <inline-formula> <tex-math notation="LaTeX">$\Theta _{h}(n^{h/2})$ </tex-math></inline-formula>. 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.