Erasure Codes for Distributed Storage: Tight Bounds and Matching Constructions

TL;DR

This thesis advances the theoretical understanding of erasure codes for distributed storage by establishing tight bounds and constructing optimal codes for regenerating and locally recoverable codes, improving efficiency and reliability.

Contribution

It characterizes the optimal rate for locally recoverable codes with sequential erasure recovery and develops tight bounds on sub-packetization for optimal-access regenerating codes.

Findings

Derived tight bounds on code rate and minimum distance.

Developed constructions for low field size Maximal Recoverable codes.

Established the optimal sub-packetization level for certain regenerating codes.

Abstract

This thesis makes several significant contributions to the theory of both Regenerating (RG) and Locally Recoverable (LR) codes. The two principal contributions are characterizing the optimal rate of an LR code designed to recover from $t$ erased symbols sequentially, for any $t$ and the development of a tight bound on the sub-packetization level (length of a vector code symbol) of a sub-class of RG codes called optimal-access RG codes. There are however, several other notable contributions as well such as deriving the tightest-known bounds on the performance metrics such as minimum distance and rate of a sub-class of LR codes known as availability codes. The thesis also presents some low field size constructions of Maximal Recoverable codes.