Abstract
A (q,m,k) cover of V(k,q), the vector space of k-tuples over GF(q), is a subset of the non-zero vectors of V(k,q) which has rank k and has non-empty intersection with every subspace of V(k,q) of rank k−m. A (q,m,k) cover may also be viewed as a matroid. As such it is essentially the image in V(k,q) of a restriction M of PG(k−1,q) under some representation, where M has rank k and critical exponent greater than m. An earlier paper answered several questions of Mullin and Stanton concerning (2,m,k) covers. This paper answers the corresponding questions for (q,m,k) covers when q>2. In particular, the least number η(q,m,k) of elements in a (q,m,k) cover is determined and those matroids having exactly η(q,m,k) elements are characterized.
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© 1979 Springer-Verlag
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Oxley, J.G. (1979). A generalization of a covering problem of mullin and stanton for matroids. In: Horadam, A.F., Wallis, W.D. (eds) Combinatorial Mathematics VI. Lecture Notes in Mathematics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0102687
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DOI: https://doi.org/10.1007/BFb0102687
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