Inverse-free Berlekamp-Massey-Sakata Algorithm and Small Decoders for Algebraic-Geometric Codes

TL;DR

This paper introduces an inverse-free decoding algorithm for algebraic-geometric codes that eliminates finite field divisions, enabling efficient decoding without syndrome calculation, and proposes three hardware architectures with performance analysis.

Contribution

The paper presents a novel inverse-free Berlekamp-Massey-Sakata algorithm and three hardware architectures for decoding algebraic-geometric codes, reducing computational complexity.

Findings

Algorithm achieves full error correction performance for generic errors.

Hardware architectures show reduced running time and resource usage.

Performance comparisons demonstrate efficiency over conventional methods.

Abstract

This paper proposes a novel algorithm for finding error-locators of algebraic-geometric codes that can eliminate the division-calculations of finite fields from the Berlekamp-Massey-Sakata algorithm. This inverse-free algorithm provides full performance in correcting a certain class of errors, generic errors, which includes most errors, and can decode codes on algebraic curves without the determination of unknown syndromes. Moreover, we propose three different kinds of architectures that our algorithm can be applied to, and we represent the control operation of shift-registers and switches at each clock-timing with numerical simulations. We estimate the performance in comparison of the total running time and the numbers of multipliers and shift-registers in three architectures with those of the conventional ones for codes on algebraic curves.