University of Illinois Urbana-Champaign

On Dihedral Codes and the Double Circulant Conjecture for Binary Extended Quadratic Residue Codes

Musa, Mona Barakat

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Description

Title
On Dihedral Codes and the Double Circulant Conjecture for Binary Extended Quadratic Residue Codes
Author(s)
Musa, Mona Barakat
Issue Date
2004
Doctoral Committee Chair(s)
Boston, Nigel
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
Graduation Semester
2004
Type of Resource
text
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http://hdl.handle.net/2142/86832

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