Najme Sahami

Topic

Double Circulant Self-Dual Codes from Legendre Sequences

Department of Electrical and Computer Engineering

Date & location

• Friday, August 9, 2024

• 1:00 P.M.

• Virtual Defence

Reviewers

Supervisory Committee

• Dr. T. Aaron Gulliver, Department of Electrical and Computer Engineering, University of Victoria (Supervisor)

• Dr. Majid Mazrooei, Department of Electrical and Computer Engineering, UVic (Member) 

External Examiner

• Dr. Peter Dukes, Department of Mathematics and Statistics, University of Victoria 

Chair of Oral Examination

• Dr. Daniela Constantinescu, Department of Mathematics and Statistics, UVic 

Abstract

A Legendre sequence s of length p, where p is an odd prime, is used to create a circulant matrix S. An alternative Legendre sequence, ˜ s, is employed to form another circulant matrix ˜ S. By concatenating these two matrices, we obtain the matrix D′, which is subsequently used to form a bordered double circulant code with length 2p + 2 and dimension k = p + 1 over GF(q), q is a prime and gcd(p,q) = 1. We demonstrate that for p = 2qm − 1 the code generated by D = 11 1T|1T 10 S|˜ S over GF(q) is self-dual. We introduce the decomposition of these codes, emphasizing their self-dual properties. Theoretical proofs are provided to support the orthogonality and self-orthogonality of the rows of these codes. Additionally, we discuss the rank of the circulant matrices formed by Legendre sequences of length p = 4kq−1 over GF(q). We demonstrate that specific row-column permutations in D′ lead to non-singular matrices, revealing that these codes can be defined as direct sums of codes generated by S and ˜ S.