| Abstrakt: |
Let p be ak-net of ordern with line-point incidence matrixN and letA be the adjacency matrix of its collinearity graph. In this paper we study thep-ranks (that is, the rank over  $$\mathbb{F}_p $$ ) of the matrixA+kl withp a prime dividingn. SinceA+kI=NTN thesep-ranks are closely related to thep-ranks ofN. Using results of Moorhouse on thep-ranks ofN, we can determinerp(A+kI) if p is a 3-net (latin square) or a desarguesian net of prime order. On the other hand we show how results for thep-ranks ofA+kI can be used to get results for thep-ranks ofN, especially in connection with the Moorhouse conjecture. Finally we generalize the result of Moorhouse on thep-rank ofN for desarguesian nets of orderp a bit to special subnets of the desarguesian affine plane of orderpe. |
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