Two Families of Optimal Linear Codes and Their Subfield Codes

In this paper, a family of [q(2) - 1, 4, q(2) - q - 2] cyclic codes over F-q meeting the Griesmer bound is presented. Their duals are [q(2) - 1, q(2) - 5, 4] almost MDS codes and are optimal with respect to the sphere-packing bound. The q(0)-ary subfield codes of this family of cyclic codes are also investigated, where q(0) is any prime power such that q is power of q(0). 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 [q(2), 4, q(2) - q - 1] linear codes over Fq meeting the Griesmer bound is presented. Their duals are [q(2), q(2) - 4, 4] almost MDS codes and are optimal with respect to the sphere-packing bound. The q(0)-ary subfield codes of this family of linear codes are also investigated, where q(0) is any prime power such that q is power of q(0). Five infinite families of 2-designs are also constructed with three families of linear codes of this paper.