## Denniston Codes

For *b* = 2^{u} a linear [2^{i}*b* + 2^{i} – *b*, 3, 2^{i}*b*−*b*]-code over **F**_{b} can be constructed for all 0 < *i* < *u*.

### Construction

Let *a* ∈ **F**_{b} such that the absolute trace of *a* is 1. Let *E* ⊂ **F**_{b} denote an *i*-dimensional **F**_{2}-vector space. The corresponding Denniston arc [1] in the projective plane PG(2, *b*) is defined as

*i*,

*b*) := {(

*X*:

*Y*: 1) :

*X*

^{2}+

*XY*+

*aY*

^{2}∈

*E*}.

The Denniston arc D(*i*, *b*) is a plane (*s*, *r*)-arc with *s* = 2^{i}*b* + 2^{i}−*b* and *r* = 2^{i}, i.e., it has *s* = 2^{i}*b* + 2^{i}−*b* points and no more than *r* = 2^{i} of its points are collinear.

The points of D(*i*, *b*) form the generator matrix of a projective [*s*, 3, *s*−*r*]-code over **F**_{b}, 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: 2008-04-04.
http://mint.sbg.ac.at/desc_CDenniston.html