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

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)
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