## Codes from Dual Arcs

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

### Construction

Let *u* := 2*b*−*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 *g*_{1},…, *g*_{u} ⊂ 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

*χ*

_{gj}(

*)*

**x**times as a column, with *χ*_{gj} denoting the characteristic function of *g*_{j}.

### Optimality

Codes from dual arcs meet the Griesmer bound with equality.

### See Also

[1, pages 255–257]

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