Improved Maximally Recoverable LRCs Using Skew Polynomials

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