A Rate-Optimal Construction of Codes with Sequential Recovery with Low Block Length

TL;DR

This paper introduces a new, rate-optimal erasure code construction with sequential recovery that significantly reduces block length compared to previous methods, using recursive graph-based techniques.

Contribution

The authors present a novel construction of rate-optimal codes with sequential recovery that achieves smaller block lengths for any t and r ≥ 3, improving upon earlier large-length codes.

Findings

Block length reduced to approximately r^{(5t/4)+(7/4)}

Construction based on recursive tree-like graphs with girth t+1

Achieves rate optimality for all t and r ≥ 3

Abstract

An erasure code is said to be a code with sequential recovery with parameters $r$ and $t$, if for any $s \leq t$ erased code symbols, there is an $s$-step recovery process in which at each step we recover exactly one erased code symbol by contacting at most $r$ other code symbols. In earlier work by the same authors, presented at ISIT 2017, we had given a construction for binary codes with sequential recovery from $t$ erasures, with locality parameter $r$, which were optimal in terms of code rate for given $r,t$, but where the block length was large, on the order of $r^{c^t}$, for some constant $c >1$. In the present paper, we present an alternative construction of a rate-optimal code for any value of $t$ and any $r\geq3$, where the block length is significantly smaller, on the order of $r^{\frac{5t}{4}+\frac{7}{4}}$ (in some instances of order $r^{\frac{3t}{2}+2}$). Our construction is based on the construction of certain kind of tree-like graphs with girth $t+1$. We construct these graphs and hence the codes recursively.