Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes

TL;DR

This paper introduces a lemma connecting linear feedback shift registers and DFTs to affine variety codes, simplifying encoding and decoding processes, and reducing computational complexity in error evaluation.

Contribution

It presents a novel lemma that unifies encoding and decoding of affine variety codes using linear maps, improving efficiency and understanding.

Findings

Error-value estimation is achieved via a new lemma.

Computational complexity is reduced from O(n^3) to O(qn^2).

Systematic encoding is shown as a special case of erasure-only decoding.

Abstract

In this paper, we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes, algebraic geometry codes, and affine variety codes. Our lemma corresponds to the non-systematic encoding of affine variety codes, and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a Grobner basis and a generalized inverse discrete Fourier transform. We clarify that our lemma yields the error-value estimation in the fast erasure-and-error decoding of a class of dual affine variety codes. Moreover, we show that systematic encoding corresponds to a special case of erasure-only decoding. The lemma enables us to reduce the computational complexity of error-evaluation from O(n^3) using Gaussian elimination to O(qn^2) with some mild conditions on n and q, where n is the code length and q is the finite-field size.