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

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Musa, Mona Barakat

Contributors dc:contributor

• Boston, Nigel

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3130989

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86832