An incidence geometry is, briefly described, a set of elements that have a different type and an incidence relation satisfying certain axioms. Typical examples include projective spaces, affine spaces, and polar spaces. The elements of type 1 are commonly called points, elements of type 2 are lines, and so on. When dealing with finite incidence geometries, that is, the number of points, lines, planes, and so on, is finite, it makes sense to consider the incidence matrix defined as follows. Consider elements of two different types, for example, points and lines, and label them. Define a matrix aij as follows: aij = 1 if and only if point i is incident with line j, and aij = 0 otherwise. The incidence matrix can be interpreted as a matrix over the field of order p, p prime. A natural question is what its rank over GF(p) is (this number is called the p-rank).
In this paper, the authors show the following results: (1) the 2-rank of the incidence matrix of points and lines of an h-dimensional projective space over GF(4) equals ; and (2) the 2-rank of the incidence matrix of points of lines of an h-dimensional affine space over GF(4) equals 4h-h2-h-1. The obtained results are related to results of Hamada on the p-rank of incidence matrices of points and higher dimensional subspaces of projective spaces.
Knowledge of (bounds on) the p-rank of an incidence matrix is particularly useful when such a matrix is used as a generator matrix or as a parity check matrix of a linear code over a field of order p. A lot of research has been done on the p-rank of incidence matrices of particular incidence structures, and the results obtained in this paper nicely contribute to this field of research.
