Sindhu Balaji

Binary codes with locality for multiple erasures having short block length

This paper considers linear, binary codes having locality parameter r, that are capable of recovering from t ≥ 2 erasures and which additionally, possess short block length. Both sequential and parallel (through orthogonal parity checks) recovery are considered. 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 high-rate constructions for (r = 2, t ∈ {4, 5, 6, 7}) and (d) providing short-block-length constructions for general (r, t). In the case of parallel repair, minimum-block-length constructions are characterized whenever t|(r2 + r) and examples examined.

On partial maximally-recoverable and maximally-recoverable codes

An [n, k] linear code C that is subject to locality constraints imposed by a parity check matrix H0 is said to be a maximally recoverable (MR) code if it can recover from any erasure pattern that some k-dimensional subcode of the null space of H0 can recover from. The focus in this paper is on MR codes constrained to have all-symbol locality r. Given that it is challenging to construct MR codes having small field size, we present results in two directions. In the first, we relax the MR constraint and require only that apart from the requirement of being an optimum all-symbol locality code, the code must yield an MDS code when punctured in a single, specific pattern which ensures that each local code is punctured in precisely one coordinate and that no two local codes share the same punctured coordinate. We term these codes as partially maximally recoverable (PMR) codes. We provide a simple construction for high-rate PMR codes and then provide a general, promising approach that needs further investigation. In the second direction, we present three constructions of MR codes with improved parameters, primarily the size of the finite field employed in the construction.

Bounds on the rate and minimum distance of codes with availability

In this paper we investigate bounds on rate and minimum distance of codes with t availability. We present bounds on minimum distance of a code with t availability that are tighter than existing bounds. For bounds on rate of a code with t availability, we restrict ourselves to a sub-class of codes with t availability called codes with strict t availability and derive a tighter rate bound. Codes with strict t availability can be defined as the null space of an (m × n) parity-check matrix H, where each row has weight (r + 1) and each column has weight t, with intersection between support of any two rows at most one. We also present two general constructions for codes with t availability.

A tight rate bound and a matching construction for locally recoverable codes with sequential recovery from any number of multiple erasures

An [n, fc] code C is said to be locally recoverable in the presence of a single erasure, and with locality parameter r, if each of the n code symbols of C can be recovered by accessing at most r other code symbols. An [n, k] code is said to be a locally recoverable code with sequential recovery from t erasures, if for any set of s ≤ t erasures, there is an s-step sequential recovery process, in which at each step, a single erased symbol is recovered by accessing at most r other code symbols. This is equivalent to the requirement that for any set of s ≤ t erasures, the dual code contain a codeword whose support contains the coordinate of precisely one of the s erased symbols. In this paper, a tight upper bound on the rate of such a code, for any value of number of erasures t and any value r ≥ 3, of the locality parameter is derived. This bound proves an earlier conjecture due to Song, Cai and Yuen. While the bound is valid irrespective of the field over which the code is defined, a matching construction of binary codes that are rate-optimal is also provided, again for any value of t and any value r ≥ 3.

Erasure coding for distributed storage: an overview

In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing the amount of repair data transferred over the network, the amount of data accessed at a helper node as well as the number of helper nodes contacted. Coding theory has evolved to handle these challenges by introducing two new classes of erasure codes, namely regenerating codes and locally recoverable codes as well as by coming up with novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an overview of the efforts in this direction that have taken place over the past decade.