The Subfield Codes of Ovoid Codes

Ovoids in ${\mathrm {PG}}(3, {\mathrm {GF}}(q))$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families of ovoids are known. The first is the elliptic quadrics and the second is the Tits ovoids. It is known that an ovoid in ${\mathrm {PG}}(3, {\mathrm {GF}}(q))$ corresponds to a $[q^{2}+1, 4, q^{2}-q]$ code over ${\mathrm {GF}}(q)$ , which is called an ovoid code. The objectives of this paper are to develop the general theories of subfield codes and to study the subfield codes of the two families of ovoid codes. The dimensions, minimum weights, and the weight distributions of the subfield codes of the elliptic quadric codes and Tits ovoid codes are settled. The parameters of the duals of these subfield codes are also studied. Some of the codes presented in this paper are optimal, and some are distance-optimal. The parameters of the subfield codes are new.