Variant Codes Based on A Special Polynomial Ring and Their Fast Computations

TL;DR

This work develops a unifying approach to binary MDS array codes by mapping parity-check matrices from the polynomial ring $\\mathbb{F}_2[x]/\\langle \\sum_{i=0}^{p-1}x^{i\\tau}\\rangle$ to binary form, yielding two new classes, V-ETBR and V-ESIP. It establishes conditions under which these variant codes inherit the MDS property from their ring-based counterparts, and provides explicit Vandermonde- and Cauchy-based constructions that support any number of parity columns and exponentially many data columns. Importantly, the paper introduces two fast syndrome computations whose asymptotic complexity per data bit is $\\lfloor\\log_2 r\\rfloor+1$, matching the best known bounds for MDS codes. The practical impact is the creation of scalable, implementable binary MDS array codes with flexible design parameters and improved encoding/decoding efficiency for large storage systems.

Abstract

Binary array codes are widely used in storage systems to prevent data loss, such as the Redundant Array of Independent Disks~(RAID). Most designs for such codes, such as Blaum-Roth~(BR) codes and Independent-Parity~(IP) codes, are carried out on the polynomial ring F_2[x]/<\sum_{i=0}^{p-1}x^i >, where F_2 is a binary field, and p is a prime number. In this paper, we consider the polynomial ring F_2[x]/<\sum_{i=0}^{p-1}x^{iτ}>, where p>1 is an odd number and τ\geq 1 is any power of two, and explore variant codes from codes over this polynomial ring. Particularly, the variant codes are derived by mapping parity-check matrices over the polynomial ring to binary parity-check matrices. Specifically, we first propose two classes of variant codes, termed V-ETBR and V-ESIP codes. To make these variant codes binary maximum distance separable~(MDS) array codes that achieve optimal storage efficiency, this paper then derives the connections between them and their counterparts over polynomial rings. These connections are general, making it easy to construct variant MDS array codes from various forms of matrices over polynomial rings. Subsequently, some instances are explicitly constructed based on Cauchy and Vandermonde matrices. In the proposed constructions, both V-ETBR and V-ESIP MDS array codes can have any number of parity columns and have the total number of data columns of exponential order with respect to $p$. In terms of computation, two fast syndrome computations are proposed for the Vandermonde-based V-ETBR and V-ESIP MDS array codes, both meeting the lowest known asymptotic complexity among MDS codes. Due to the fact that all variant codes are constructed from parity-check matrices over simple binary fields instead of polynomial rings, they are attractive in practice.

Figures (4)

• Figure 1: Diagram of the BR code with $p=5$ and $r=3$.
• Figure 2: Diagram of the generalized RDP code with $p=5$ and $r=3$.
• Figure 3: Computational complexities of different binary MDS array codes (when the total number of data columns is $127$).
• Figure 4: Computational complexities of different binary MDS array codes (when the total number of data columns is $251$).

Theorems & Definitions (30)

• Theorem 1
• Definition 1
• Remark 1
• Definition 2
• Remark 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2