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On the \(p\)-rank of incidence matrices and a bound of Bruen and Ott. (English) Zbl 0772.51008

The authors discuss the following problem: what is a lower bound of the rank \(rk_ p(B)\) of the incidence matrix \(B\) of a finite partial linear space (PLS) viewed as a matrix over a field with characteristic \(p\). The first bound they obtain results in investigating the flag matrix \(F\) of a given PLS. The authors prove that for a PLS with \(\nu\) points, \(b\) lines and \(f\) flags, and degree of any point and line congruent to 1 modulo \(p\) we have \((rk_ p(B)-1)^ 2 \geq f - \nu - b + 1\). For projective planes of order \(n\) with \(p\) dividing \(n\), this bound coincides with the bound obtained by Bruen and Ott. To obtain the second bound the authors use techniques of incidence graphs. Then they prove that for a connected PLS with, \(\nu\), \(b\) and \(f\) as above it holds \(rk_ p(B)^ 2 \geq f - \nu - b + 1\). At the end of the note the authors give explicit computer results for \(rk_ p(B)\) and \(rk_ p(F)\) of some projective planes of small order, with small \(p\).

MSC:

51E14 Finite partial geometries (general), nets, partial spreads
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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References:

[1] N.L. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge (1974). · Zbl 0284.05101
[2] A.A. Bruen and U. Ott, On the p-rank of incidence matrices and a question of E.S. Lander, in : Finite Geometries and Combinatorial Designs, (eds. E.S. Kramer and S. Magliveras) American Math. Soc., (1990). · Zbl 0712.05014
[3] G. Hillbrandt, The p-rank of 01-matrices, J. Combinatorial Theory, Series A Vol. 60 (1992) pp. 131-139. · Zbl 0757.05034 · doi:10.1016/0097-3165(92)90043-T
[4] E.S. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press, Cambridge (1983). · Zbl 0502.05010
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