Denniston Codes
For b = 2u a linear [2ib + 2i – b, 3, 2ib−b]-code over Fb can be constructed for all 0 < i < u.
Construction
Let a ∈ Fb such that the absolute trace of a is 1. Let E ⊂ Fb denote an i-dimensional F2-vector space. The corresponding Denniston arc [1] in the projective plane PG(2, b) is defined as
The Denniston arc D(i, b) is a plane (s, r)-arc with s = 2ib + 2i−b and r = 2i, i.e., it has s = 2ib + 2i−b points and no more than r = 2i of its points are collinear.
The points of D(i, b) form the generator matrix of a projective [s, 3, s−r]-code over Fb, the Denniston code. Denniston codes are two-weight codes. The only non-zero weights are s and s−r.
Optimality
Denniston codes meet the Griesmer bound with equality.
See Also
References
| [1] | Ralph H. F. Denniston. Some maximal arcs in finite projective planes. Journal of Combinatorial Theory, 6:317–319, 1969. |
| [2] | Jürgen Bierbrauer. Introduction to Coding Theory. Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004. MR2079734 (2005f:94001) |
Copyright
Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Denniston Codes.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_CDenniston.html