Advances in Mathematics of Communications (AMC)

Double circulant codes from two class association schemes

Pages: 45 - 64, Volume 1, Issue 1, February 2007      doi:10.3934/amc.2007.1.45

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Steven T. Dougherty - Department of Mathematics, University of Scranton, Scranton, PA 18518, United States (email)
Jon-Lark Kim - Department of Mathematics, University of Louisville, Louisvile, KY 40292, United States (email)
Patrick Solé - CNRS, I3S, ESSI, BP 145, Route des Colles, 06 903 Sophia Antipolis, France (email)

Abstract: Two class association schemes consist of either strongly regular graphs (SRG) or doubly regular tournaments (DRT). We construct self-dual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and Gaborit Quadratic Double Circulant codes (QDC). As new examples SRG's give 4 (resp. 5) new Type I (resp. Type II) [72, 36, 12] codes. We construct a [200, 100, 12] Type II code invariant under the Higman-Sims group, a [200, 100, 16] Type II code invariant under the Hall-Janko group, and more generally self-dual binary codes attached to rank three groups.

Keywords:  Self-dual code, 2-class association scheme, strongly regular graph, rank three groups, doubly regular tournament.
Mathematics Subject Classification:  Primary: 94B60; Secondary: 05E30.

Received: April 2006;      Revised: August 2006;      Available Online: January 2007.