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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 UrbanaChampaign 
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 UrbanaChampaign, 2004. 
URI:  http://hdl.handle.net/2142/86832 
Other Identifier(s):  (MiAaPQ)AAI3130989 
Date Available in IDEALS:  20150928 
Date Deposited:  2004 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois