The fast decoding of Reed-Solomon codes using Fermat transforms (Corresp.)

Author

Reed, I. ; Treiu-Kien Truong ; Welch, L.

Volume

Issue

fYear

1978

fDate

7/1/1978 12:00:00 AM

Firstpage

497

Lastpage

499

Abstract

It is shown that $\\sqrt [8]{2}$ is an element of order $2^{n+4}$ in $GF(F_{n})$ , where $F_{n}=2^{2^{n}}+1$ is a Fermat prime for $n=3,4$ . Hence it can be used to define a fast Fourier transform (FFT) of as many as $2^{n+4}$ symbols in $GF(F_{n})$ . Since $\\sqrt [8]{2}$ is a root of unity of order $2^{n+4}$ in $GF(F_{n})$ , this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the case $n=2$ , where $\\sqrt {2}$ is in $GF(F_{2})=GF(17)$ , is given.

Keywords

Decoding; Number-theoretic transforms; Reed-Solomon codes; Contracts; Decoding; Error correction codes; Fast Fourier transforms; Galois fields; Information science; Linear code; Mathematics; Probes; Reed-Solomon codes;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.1978.1055902

Filename

1055902

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=928707