Denniston Codes

For b = 2u a linear [2ib + 2ib, 3, 2ibb]-code over Fb can be constructed for all 0 < i < u.

Construction

Let aFb such that the absolute trace of a is 1. Let EFb denote an i-dimensional F2-vector space. The corresponding Denniston arc [1] in the projective plane PG(2, b) is defined as

D(i, b) := {(X : Y : 1)  :  X2 + XY + aY2E}.

The Denniston arc D(i, b) is a plane (s, r)-arc with s = 2ib + 2ib and r = 2i, i.e., it has s = 2ib + 2ib 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, sr]-code over Fb, the Denniston code. Denniston codes are two-weight codes. The only non-zero weights are s and sr.

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: 2008-04-04. http://mint.sbg.ac.at/desc_CDenniston.html

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