Binary Codes with Locality for Multiple Erasures Having Short Block Length

TL;DR

This paper investigates binary linear codes with locality that can recover from multiple erasures, focusing on short block lengths, and provides new constructions and bounds for both sequential and parallel recovery methods.

Contribution

It introduces new constructions and bounds for short block length binary codes capable of multiple erasure recovery with locality, extending previous results for various parameters.

Findings

Minimum block length constructions for parallel repair for general t.

Extended and characterized minimum block length for t=2.

Improved bounds and constructions for t=3.

Abstract

The focus of this paper is on linear, binary codes with locality having locality parameter $r$, that are capable of recovering from $t\geq 2$ erasures and that moreover, have short block length. Both sequential and parallel (through orthogonal parity checks) recovery is considered here. In the case of parallel repair, minimum-block-length constructions for general $t$ are discussed. In the case of sequential repair, the results include (a) extending and characterizing minimum-block-length constructions for $t=2$, (b) providing improved bounds on block length for $t=3$ as well as a general construction for $t=3$ having short block length, (c) providing short-block-length constructions for general $r,t$ and (d) providing high-rate constructions for $r=2$ and $t$ in the range $4 \leq t \leq7$. Most of the constructions provided are of binary codes.