On decoding of quadratic residue codes

Dissertant

al-Shorbassi, Ramiz Zahar

Thesis advisor

al-Ashker, Muhammad Mahmud

University

Islamic University

Faculty

Faculty of Science

Department

Department of Mathematics

University Country

Palestine (Gaza Strip)

Degree

Master

Degree Date

2010

English Abstract

A binary Quadratic Residue (QR) code of length n is an (n, (n+1) / 2) cyclic code over GF (2m) with generator polynomial g(x) where m is some integer.

The length of this code is a prime number of the form n = 8l + 1 where l is some integer.

The generator polynomial g(x) is defined by g(?) Π i∈Qn (? – βi) where β is a primitive nth root of unity in the finite field GF (2m) with m being the smallest positive integer such that n|2m−1 and Qn is the collection of all nonzero quadratic residues modulo n given by Qn = {i|i ≡ j2 mod n for 1 ≤ j ≤ n – 1}. Algebraic approaches to the decoding of the quadratic residue (QR) codes were studied in [2], [3], [4], [5], [6] and [13].

Here, in this thesis, some new more general properties are found for the syndromes of the subclass of binary QR codes of length n = 8m + 1 or n = 8m – 1.

A new algebraic decoding algorithm for the (41, 21, 9) binary QR code is presented by having the unknown syndrome S3 which is a necessary condition for decoding the (41, 21, 9) QR code.

Main Subjects

Mathematics

Topics

• Algorithms
• Algebraic number theory

No. of Pages

87

Table of Contents

• APA
• MLA
• AMA

Language

English

Data Type

Arab Theses

Record ID

BIM-300284