Two Families of Optimal Linear Codes and Their Subfield Codes

In this paper, a family of [q2- 1, 4, q2- q - 2] cyclic codes over Fqmeeting the Griesmer bound is presented. Their duals are [q2- 1, q2- 5,4] almost MDS codes and are optimal with respect to the sphere-packing bound. The q0-ary subfield codes of this family of cyclic codes are also investigated, where q0is any prime power such that q is power of q0. Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of [q2, 4, q2- q - 1] linear codes over Fqmeeting the Griesmer bound is presented. Their duals are [q2, q2- 4, 4] almost MDS codes and are optimal with respect to the sphere-packing bound. The q0-ary subfield codes of this family of linear codes are also investigated, where q0is any prime power such that q is power of q0. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper.