Caps and codes

Abstract 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.