Construction of quantum caps in projective space PG(r, 4) and quantum codes of distance 4

Abstract

Constructions of quantum caps in projective space PG(r, 4) by recursive methods and computer search are discussed. For each even n satisfying $$n\ge 282$$n?282 and each odd z satisfying $$z\ge 275$$z?275, a quantum n-cap and a quantum z-cap in $$PG(k-1, 4)$$PG(k-1,4) with suitable k are constructed, and $$[[n,n-2k,4]]$$[[n,n-2k,4]] and $$[[z,z-2k,4]]$$[[z,z-2k,4]] quantum codes are derived from the constructed quantum n-cap and z-cap, respectively. For $$n\ge 282$$n?282 and $$n\ne 286$$n?286, 756 and 5040, or $$z\ge 275$$z?275, the results on the sizes of quantum caps and quantum codes are new, and all the obtained quantum codes are optimal codes according to the quantum Hamming bound. While constructing quantum caps, we also obtain many large caps in PG(r, 4) for $$r\ge 11$$r?11. These results concerning large caps provide improved lower bounds on the maximal sizes of caps in PG(r, 4) for $$r\ge 11$$r?11.