Caps and codes

The packing problem in the theory of caps is that of finding, or at least bounding, the size m(r, q) of an ovaloid (cap of largest size) in the projective space S_r,q of dimension r over a field of q elements. This problem and that of constructing and classifying ovaloids are approached by consideration of certain codes associated with caps. Improved general upper bounds on m(r, q) are found, which give m(5, 3)?56 as a particular case. A 56-cap in S_5,3 is constructed via its code and its uniqueness as an ovaloid is demonstrated.