基于特殊多项式环的变体码及其快速计算
Variant Codes Based on A Special Polynomial Ring and Their Fast Computations
AI总结：
本文在特殊多项式环F_2[x]/<\sum_{i=0}^{p-1}x^{iτ}>上提出V-ETBR和V-ESIP两类变体码，通过将多项式环奇偶校验矩阵映射为二进制矩阵构造MDS阵列码，并给出基于Vandermonde矩阵的快速伴随式计算，达到已知最低渐近复杂度。
摘要：
二进制阵列码广泛用于存储系统中以防止数据丢失，例如独立磁盘冗余阵列（RAID）。此类码的大多数设计，如Blaum-Roth（BR）码和Independent-Parity（IP）码，都是在多项式环F_2[x]/<\sum_{i=0}^{p-1}x^i>上进行的，其中F_2是二进制域，p是素数。本文考虑多项式环F_2[x]/<\sum_{i=0}^{p-1}x^{iτ}>，其中p>1是奇数，τ≥1是任意2的幂，并从该多项式环上的码探索变体码。特别地，变体码是通过将多项式环上的奇偶校验矩阵映射为二进制奇偶校验矩阵而得到的。具体而言，我们首先提出两类变体码，称为V-ETBR和V-ESIP码。为了使这些变体码成为实现最优存储效率的二进制最大距离可分（MDS）阵列码，本文随后推导了它们与多项式环上对应码之间的联系。这些联系具有一般性，使得从多项式环上各种形式的矩阵构造变体MDS阵列码变得容易。随后，基于Cauchy矩阵和Vandermonde矩阵显式构造了一些实例。在所提出的构造中，V-ETBR和V-ESIP MDS阵列码都可以具有任意数量的奇偶校验列，并且数据列总数相对于p呈指数阶。在计算方面，针对基于Vandermonde的V-ETBR和V-ESIP MDS阵列码提出了两种快速伴随式计算，两者都达到了MDS码中已知最低的渐近复杂度。由于所有变体码都是从简单二进制域上的奇偶校验矩阵而非多项式环构造的，因此它们在实际中具有吸引力。
英文摘要：
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.
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