On complete caps in the projective geometries over $$\mathbb{F}_3 $$

Abstract

It is known that the largest size of cap in PG(5, 3) is 56, but very little is known about complete caps of smaller size; the previously known complete caps withk < 56 all had size at most 43. In this paper we construct complete 48-caps and show that any 53-cap is extendable to a 56-cap. From this last result, we derive new upper bounds on the largest size of cap in PG(r, 3) forr ≥ 6. The results are obtained from a blend of geometric and coding theoretic techniques.