Encoding via Gr\"obner bases and discrete Fourier transforms for several types of algebraic codes
Hajime Matsui, Seiichi Mita
TL;DR
This paper introduces a new encoding method for algebraic codes using Gr"obner bases and discrete Fourier transforms, enabling systematic encoding for various complex algebraic codes with demonstrated numerical examples.
Contribution
It presents a novel encoding scheme that generalizes generator polynomial concepts and combines Gr"obner bases with Fourier transforms for multiple algebraic code types.
Findings
Effective encoding for algebraic codes demonstrated
Generalization of generator polynomial concept achieved
Numerical examples validate the approach
Abstract
We propose a novel encoding scheme for algebraic codes such as codes on algebraic curves, multidimensional cyclic codes, and hyperbolic cascaded Reed-Solomon codes and present numerical examples. We employ the recurrence from the Gr\"obner basis of the locator ideal for a set of rational points and the two-dimensional inverse discrete Fourier transform. We generalize the functioning of the generator polynomial for Reed-Solomon codes and develop systematic encoding for various algebraic codes.
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Taxonomy
TopicsCoding theory and cryptography · Cryptographic Implementations and Security · Cryptography and Residue Arithmetic