Sahami, Najme2024-08-272024-08-272024https://hdl.handle.net/1828/20321[see abstract in PDF of theses for correct formatting] A Legendre sequence s of length p, p an odd prime, is used to create a circulant matrix S. An alternative Legendre sequence s̃ is employed to form another circulant matrix S. By concatenating these two matrices, we obtain the matrix D′ which is used to form a bordered double circulant code with length 2p + 2 and dimension k = p + 1 over Fq, q a prime, nd gcd(p, q) = 1. We demonstrate that for p = 2qm − 1 the code generated by D=11/1/1/10/S/S over Fq is self-dual. The idempotent elements of the ideals generated by s(x) and s˜(x), the leading polynomials of the p×p matrices S and S˜, respectively, for p = 4kq−1 are investigated and used to find the rank of these matrices over Fq. We define a specific row-column permutation of [S|S˜] which leads to a non-singular matrix, revealing that these codes can be defined as a direct sum of codes.enAvailable to the World Wide WebQuadratic residue modulo pLegendre sequenceDouble circulant codesIdempotent idealSelf-dual codesDouble circulant matrixAlgebra of polynomials modulo x^p - 1Thesis