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Boston, Nigel Musa, Mona Barakat 2015-09-28T15:19:45Z 2015-09-28T15:19:45Z 10000-01-01 2004 2004 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. Made available in DSpace on 2015-09-28T15:19:45Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3130989.pdf: 3188162 bytes, checksum: 172d20d68bfa2e40184029df24a1a67d (MD5) Previous issue date: 2004 Embargo set by: Seth Robbins for item 88113 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 72 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2004. http://hdl.handle.net/2142/86832 (MiAaPQ)AAI3130989 eng Mathematics On Dihedral Codes and the Double Circulant Conjecture for Binary Extended Quadratic Residue Codes text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.