1. Introduction. In [2] by means of a fairly lengthy argument involving Hermitian varieties it was shown that PG(2n, q ) can be partitioned into (q 2n+] + !)/(# + l)-caps. Moreover, these caps were shown to constitute the "large points" of a PG(2n, q) in a natural way. In [3] a similar argument was used to show that once two disjoint (n -l)-subspaces are removed from PG(2n -1, q ), the remaining points can be partitioned into (q 2n -\)/(q 2 -l)-caps.The purpose of this paper is to give a short proof of the results found in [2], and then use the technique developed to partition PG(2n -1, q) into (cp + l)-caps for n even and q any prime-power. Moreover, these caps can be treated in a natural way as the "large points" of a PG(n -\,q). Special attention will be paid to the case n = 2, where the theorem says that PG(3, q) can be partitioned into ovoids so long as q > 2. Designs based on this ovoidal fibration will then be constructed. PG(d, q) denote the desarguesian tridimensional projective geometry over the finite field GF(q). A k-cap in 2 is a collection of k points in 2 with no three collinear. We will represent 2 in the following way. Since the finite field GF(cf +x ) is a (d + 1)dimensional vector space over GF(q), the points of 2 can be thought of as the 1-dimensional subspaces of this vector space. If ft is a primitive element of GF(^+ 1 ) and N = (cf +x -\)/(q -1), then fi N is a primitive element of the subfield GF{q). Hence the points of 2 can be identified with the field elements /? 6 , p\ /?, . .., p N~K If 0 ^ i : ¥= j : ^ N -1, the line L joining /?' and P J will consist of the points Background. Let 2 =where fi 4-a/3 J is computed as some P k in the field GF(^^1) and then the exponent k is read modulo N.3. Partitioning PG(2n, q 2 ). Theorem (1) below contains the results found in [2], although the proof is quite different.

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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

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