Najme Sahami
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BSc (Kourdestan University, 2003)
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MSc (University of Yazd, 2007)
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PhD (University of Kashan, 2020)
Topic
Double Circulant Self-Dual Codes from Legendre Sequences
Department of Electrical and Computer Engineering
Date & location
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Friday, August 9, 2024
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1:00 P.M.
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Virtual Defence
Reviewers
Supervisory Committee
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Dr. T. Aaron Gulliver, Department of Electrical and Computer Engineering, 番茄社区 (Supervisor)
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Dr. Majid Mazrooei, Department of Electrical and Computer Engineering, UVic (Member)
External Examiner
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Dr. Peter Dukes, Department of Mathematics and Statistics, 番茄社区
Chair of Oral Examination
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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.