Dual codes of projective planes of order 25. (English) Zbl 1038.51007

Clark, K. L.; Hatfield, L. D.; Key, J. D.; Ward, H. N.

Adv. Geom. 2003, Spec. Issue, S140-S152 (2003).

The authors study the minimum weight of the dual linear code over GF\((5)\) of the linear code defined by the incidence matrix of the points with the lines of a projective plane of order 25.
They prove that this minimum weight \(d^\perp\) is either 42, or \(44 \leq d^\perp \leq 50\). If a Baer subplane is present in the projective plane of order 25, then \(d^\perp\) is either \(42,44\) or 45, and they also describe how the possible codewords having these minimum weights \(42,44,45\) look like.
In particular, their results show that the dual 5-ary linear code of the desarguesian projective plane PG\((2,25)\) has minimum weight 45, and that the minimum weight codewords are equal to a scalar multiple of \(v^\beta-v^l\), where \(v^\beta\) is the incidence vector of a Baer subplane \(\beta\) of PG\((2,25)\) and where \(v^l\) is the incidence vector of a line \(l\) of this projective plane of order 25 which also is a line of \(\beta\).

Reviewer: Leo Storme (Gent)

MSC:

51E15	Finite affine and projective planes (geometric aspects)
05B25	Combinatorial aspects of finite geometries
51E20	Combinatorial structures in finite projective spaces
94B05	Linear codes (general theory)

Keywords:

incidence matrices; linear codes; projective planes; minimum weight