Codes from Dual Arcs

A linear [i(b + 1) + 2, 3, ib]-code C over Fb can be constructed for all b + 1 ≤ i ≤ 2b−1.

Construction

Let u := 2b−i and choose an u-arc in the projective plane PG(2, b) (e.g., from a Reed-Solomon code). After dualizing (exchanging points with lines), a dual arc of u lines g1,…, gu ⊂ PG(2, b) is obtained such that no more than two lines intersect in a common point.

Now the points of PG(2, b) form the generator matrix of C such that every x ∈ PG(2, b) occurs exactly

2 – $\displaystyle \sum_{{j=1}}^{{u}}$χgj(x)

times as a column, with χgj denoting the characteristic function of gj.

Optimality

Codes from dual arcs meet the Griesmer bound with equality.

See Also

References

[1]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. “Codes from Dual Arcs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_CDualArc.html

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