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 Title: On Dihedral Codes and the Double Circulant Conjecture for Binary Extended Quadratic Residue Codes Author(s): Musa, Mona Barakat Doctoral Committee Chair(s): Boston, Nigel Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: Let p be a prime such that p ≡ -1 mod 8. Let k = (p + 1)/2 and write k = 2mq, q odd. Let S = F2[x]/⟨1 + xk⟩ where F2 is the Galois field of two elements. We identify the binary extended quadratic residue codes of length 2k as principal left ideals in the group algebra F2Dk where Dk is the dihedral group of order 2 k. We prove that these codes have a double circulant presentation in the following three cases: (1) q = 1. (2) q is a prime and 2 is a primitive root modulo q . (3) Let X be the class of x in S, and sigma the algebra automorphisms on S that sends Xi to X -i. Factor 1 + xq over F2 into irreducible factors. If the class of those factors in S is fixed by sigma up to a unit, then the codes have a double circulant presentation. Issue Date: 2004 Type: Text Language: English Description: 72 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2004. URI: http://hdl.handle.net/2142/86832 Other Identifier(s): (MiAaPQ)AAI3130989 Date Available in IDEALS: 2015-09-28 Date Deposited: 2004
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