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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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