Folded Algebraic Geometric Codes From Galois Extensions
Ming-Deh Huang, Anand Kumar Narayanan

TL;DR
This paper introduces a new class of list-decodable algebraic geometric codes derived from Galois extensions, extending folded Reed Solomon codes to function fields, with constructions depending on automorphism order and promising good error correction capabilities.
Contribution
It extends folded Reed Solomon codes to algebraic geometric codes using Galois automorphisms, providing new constructions with polynomial list size bounds and error correction guarantees.
Findings
Codes can correct a fraction of 1-R-ε errors with rate R.
Large automorphism order yields polynomially bounded list size.
Application to Garcia-Stichtenoth tower produces codes with specific alphabet size and error correction.
Abstract
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using certain elements (automorphisms) from the Galois group of the extension. This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes. We describe two constructions based on this framework depending on if the order of the automorphism used to fold the code is large or small compared to the block length. When the automorphism is of large order, the codes have polynomially bounded list size in the worst case. This construction gives codes of rate over an alphabet of size independent of block length that can correct a fraction of errors subject to the existence of asymptotically good towers of function…
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Taxonomy
TopicsCoding theory and cryptography · Cryptography and Residue Arithmetic
