Despite of tremendous research on decoding Reed-Solomon (RS) and algebraic geometry (AG) codes under the random and adversary substitution error models, few studies have explored these codes under the burst substitution error model. Burst errors are prevalent in many communication channels, such as wireless networks, magnetic recording systems, and flash memory. Compared to random and adversarial errors, burst errors often allow for the design of more efficient decoding algorithms. However, achieving both an optimal decoding radius and quasi-linear time complexity for burst error correction remains a significant challenge. The goal of this paper is to design (both list and probabilistic unique) decoding algorithms for RS and AG codes that achieve the Singleton bound for decoding radius while maintaining quasi-linear time complexity. Our idea is to build a one-to-one correspondence between AG codes (including RS codes) and interleaved RS codes with shorter code lengths (or even constant lengths). By decoding the interleaved RS codes with burst errors, we derive efficient decoding algorithms for RS and AG codes. For decoding interleaved RS codes with shorter code lengths, we can employ either the naive methods or existing algorithms. This one-to-one correspondence is constructed using the generalized fast Fourier transform (G-FFT) proposed by Li and Xing (SODA 2024). The G-FFT generalizes the divide-and-conquer technique from polynomials to algebraic function fields. More precisely speaking, assume that our AG code is defined over a function field $E$ which has a sequence of subfields $\mathbb{F}_q(x)=E_r\subseteq E_{r-1}\subseteq \cdots\subset E_1\subseteq E_0=E$ such that $E_{i-1}/E_i$ are Galois extensions for $1\le i\le r$. Then the AG code based on $E$ can be transformed into an interleaved RS code over the rational function field $\mathbb{F}_q(x)$.

翻译：尽管在随机和对抗替换错误模型下对Reed-Solomon（RS）码和代数几何（AG）码的译码已进行了大量研究，但在突发替换错误模型下对这些码的研究却很少。突发错误在许多通信信道中普遍存在，例如无线网络、磁记录系统和闪存。与随机错误和对抗错误相比，突发错误通常允许设计更高效的译码算法。然而，为突发纠错同时实现最优译码半径和准线性时间复杂度仍然是一个重大挑战。本文的目标是为RS码和AG码设计（包括列表译码和概率唯一译码）译码算法，使其在保持准线性时间复杂度的同时达到Singleton界的译码半径。我们的核心思想是建立AG码（包括RS码）与具有较短码长（甚至恒定码长）的交织RS码之间的一一对应关系。通过译码具有突发错误的交织RS码，我们推导出RS码和AG码的高效译码算法。对于较短码长的交织RS码译码，我们可以采用朴素方法或现有算法。这种一一对应关系是利用Li和Xing（SODA 2024）提出的广义快速傅里叶变换（G-FFT）构建的。G-FFT将分治技术从多项式推广到代数函数域。更精确地说，假设我们的AG码定义在函数域$E$上，该函数域具有一系列子域$\mathbb{F}_q(x)=E_r\subseteq E_{r-1}\subseteq \cdots\subset E_1\subseteq E_0=E$，使得对于$1\le i\le r$，$E_{i-1}/E_i$是伽罗瓦扩张。那么基于$E$的AG码可以转化为有理函数域$\mathbb{F}_q(x)$上的交织RS码。