Encoding of algebraic geometry codes with quasi-linear complexity $O(N\log N)$
Songsong Li, Shu Liu, Liming Ma, Yunqi Wan, Chaoping Xing
TL;DR
The paper tackles the problem of encoding algebraic geometry codes with quasi-linear time, introducing a divide-and-conquer FMPE framework that generalizes FFT techniques to curves via function-field towers. By enforcing a four-property condition (P4) on abelian or mix-structured extensions, it reduces MPE on a large extension to multiple MPEs on a base field, yielding overall $O(N\log N)$ encoding when a fast base-field FMPE is available. The results cover both plane-curve AG codes (including norm-trace and Hermitian-like) and non-plane constructions (Hermitian towers), with explicit instantiations using Kummer, Artin-Schreier, and mixed extensions under suitable smoothness conditions on $q$ or $q-1$. This work improves prior $O(N\log^2 N)$ or $O(N^{w/2})$ encodings and enables practical fast encoding of a broad class of AG codes for communications and data integrity applications.
Abstract
Fast encoding and decoding of codes have been always an important topic in code theory as well as complexity theory. Although encoding is easier than decoding in general, designing an encoding algorithm of codes of length $N$ with quasi-linear complexity $O(N\log N)$ is not an easy task. Despite the fact that algebraic geometry codes were discovered in the early of 1980s, encoding algorithms of algebraic geometry codes with quasi-linear complexity $O(N\log N)$ have not been found except for the simplest algebraic geometry codes--Reed-Solomon codes. The best-known encoding algorithm of algebraic geometry codes based on a class of plane curves has quasi-linear complexity at least $O(N\log^2 N)$. In this paper, we design an encoding algorithm of algebraic geometry codes with quasi-linear complexity $O(N\log N)$. Our algorithm works well for a large class of algebraic geometry codes based on both plane and non-plane curves. The main idea of this paper is to generalize the divide-and-conquer method from the fast Fourier Transform over finite fields to algebraic curves. Suppose we consider encoding of algebraic geometry codes based on an algebraic curve ${\mathcal X}$ over $\mathbb{F}_q$. We first consider a tower of Galois coverings ${\mathcal X}={\mathcal X}_0\rightarrow{\mathcal X}_1\rightarrow\cdots\rightarrow{\mathcal X}_r$ over a finite field $\mathbb{F}_q$, i.e., their function field tower $\mathbb{F}_q({\mathcal X}_0)\supsetneq\mathbb{F}_q({\mathcal X}_{1})\supsetneq\cdots \supsetneq\mathbb{F}_q({\mathcal X}_r)$ satisfies that each of extension $\mathbb{F}_q({\mathcal X}_{i-1})/\mathbb{F}_q({\mathcal X}_i)$ is a Galois extension and the extension degree $[\mathbb{F}_q({\mathcal X}_{i-1}):\mathbb{F}_q({\mathcal X}_i)]$ {is a constant}. Then encoding of an algebraic geometry code based on ${\mathcal X}$ is reduced to the encoding of an algebraic geometry code based on ${\mathcal X}_r$.