It is shown that the maximum size of a set S of vectors of a k-dimensional vector space over F q , with the property that every subset of size k is a basis, is at most q + 1 if k ≤ p, and at most q + k − p if q ≥ k ≥ p + 1 ≥ 4, where q = p h and p is prime. Moreover, for k ≤ p, the sets S of maximum size are classified, generalising Beniamino Segre's "arc is a conic" theorem.These results have various implications. One such implication is that a k × (p + 2) matrix, with k ≤ p and entries from F p , has k columns which are linearly dependent. Another is that the uniform matroid of rank r that has a base set of size n ≥ r + 2 is representable over F p if and only if n ≤ p + 1. It also implies that the main conjecture for maximum distance separable codes is true for prime fields; that there are no maximum distance separable linear codes over F p , of dimension at most p, longer than the longest Reed-Solomon codes. The classification implies that the longest maximum distance separable linear codes, whose dimension is bounded above by the characteristic of the field, are Reed-Solomon codes.In the autumn of 2008 while I was visiting Budapest, together with Andras Gács, we formulated the coordinate free version of Segre's lemma of tangents (Lemma 2.1) which is fundamental to this article. I dedicate this work to Andras, whose humour, enthusiasm and brilliance I am grateful to have known.

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Given a set U of size q in an affine plane of order q, we determine the possibilities for the number of directions of secants of U, and in many cases characterize the sets U with given number of secant directions. Academic Press

We introduce the concept of a pentagonal geometry as a generalization of the pentagon and the Desargues configuration, in the same vein that the generalized polygons share the fundamental properties of ordinary polygons. In short, a pentagonal geometry is a regular partial linear space in which for all points x, the points not collinear with the point x, form a line. We compute bounds on their parameters, give some constructions, obtain some nonexistence results for seemingly feasible parameters and suggest a cryptographic application related to identifying codes of partial linear spaces.

The projective and polar geometries that arise from a vector space over a finite field are particularly useful in the construction of combinatorial objects, such as latin squares, designs, codes and graphs. This book provides an introduction to these geometries and their many applications to other areas of combinatorics. Coverage includes a detailed treatment of the forbidden subgraph problem from a geometrical point of view, and a chapter on maximum distance separable codes, which includes a proof that such codes over prime fields are short. The author also provides more than 100 exercises (complete with detailed solutions), which show the diversity of applications of finite fields and their geometries. Finite Geometry and Combinatorial Applications is ideal for anyone, from a third-year undergraduate to a researcher, who wishes to familiarise themselves with and gain an appreciation of finite geometry.

A proof is presented that shows that the number of directions determined by a function over a finite field GF ðqÞ is either 1, at least ðq þ 3Þ=2; or between q=s þ 1 and ðq À 1Þ=ðs À 1Þ for some s where GF ðsÞ is a subfield of GF ðqÞ: Moreover, the graph of those functions that determine less than half the directions is GF ðsÞ-linear. This completes the unresolved cases s ¼ 2 and 3 of the main theorem in Blokhuis et al. (J. Combin. Theory Ser. A 86 (1999) 187).

A finite semifield is shown to be equivalent to the existence of a particular geometric configuration of subspaces with respect to a Desarguesian spread in a finite dimensional vector space over a finite field. In 1965 Knuth \cite{Knuth1965} showed that each finite semifield generates in total six (not necessarily isotopic) semifields. In certain cases, the geometric interpretation obtained here allows us to construct another six semifields, providing a link between some known examples which are not related by Knuth's operations

It is known that every ovoid of the parabolic quadric Q(4, q), q = p h , p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p = 2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points.We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q > 3.We conclude with a 1 mod p result for ovoids of Q(6, q), q = p h , p prime.