Exact-Repair Minimum Bandwidth Regenerating Codes Based on Evaluation of Linearized Polynomials

TL;DR

This paper introduces two novel constructions of exact-repair minimum bandwidth regenerating codes using linearized polynomials, enabling efficient data repair over the base field with flexible parameters.

Contribution

The paper presents a new approach for constructing exact-MBR codes using linearized polynomials evaluated over extension fields, a method not previously explored in this context.

Findings

Allows data repair over GF(q) without matrix inversion

Supports arbitrary parameters for exact-MBR codes

Offers lower repair complexity but higher reconstruction complexity

Abstract

In this paper, we propose two new constructions of exact-repair minimum storage regenerating (exact-MBR) codes. Both constructions obtain the encoded symbols by first treating the message vector over GF(q) as a linearized polynomial and then evaluating it over an extension field GF(q^m). The evaluation points are chosen so that the encoded symbols at any node are conjugates of each other, while corresponding symbols of different nodes are linearly dependent with respect to GF(q). These properties ensure that data repair can be carried out over the base field GF(q), instead of matrix inversion over the extension field required by some existing exact-MBR codes. To the best of our knowledge, this approach is novel in the construction of exact-MBR codes. One of our constructions leads to exact-MBR codes with arbitrary parameters. These exact-MBR codes have higher data reconstruction complexities but lower data repair complexities than their counterparts based on the product-matrix approach; hence they may be suitable for applications that need a small number of data reconstructions but a large number of data repairs.