Encoding of algebraic geometry codes with quasi-linear complexity $O(N\log N)$

TL;DR

This paper introduces a novel encoding algorithm for algebraic geometry codes that achieves quasi-linear complexity $O(N ext{log}N)$, significantly improving over previous methods and applicable to a broad class of such codes.

Contribution

It generalizes the divide-and-conquer approach from FFT to algebraic curves, enabling efficient encoding of a wide range of algebraic geometry codes.

Findings

Achieves $O(N ext{log}N)$ encoding complexity for algebraic geometry codes.

Applicable to codes based on both plane and non-plane algebraic curves.

Extends the divide-and-conquer method from FFT to algebraic geometry code encoding.

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 ${\mathcal X}_r$.