In this paper, a family of [q2-1,4,q2 -q-2] cyclic codes over 픽q meeting 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 q0 is 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 픽q meeting 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 q0 is 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.

Codebooks are required to have small inner-product correlation in many practical applications, such as direct spread code division multiple access communications, space-time codes and compressed sensing. In general, it is difficult to construct optimal codebooks. In this paper, two kinds of codebooks are presented and proved to be optimally optimal with respect to the Welch bound. Additionally, the constructed codebooks in this paper have new parameters.

Codebooks with low coherence have extensive applications in many fileds such as code division multiple access (CDMA) communication systems, MIMO communications, compressed sensing and so on. In this paper, based on additive characters over finite fields, we propose a construction of optimal codebook with respect to the Levenshtein bound and verify that it is a new construction. By shortening the length of the optimal codebooks, we present a construction of codebooks asymptotically meeting the Levenshtein bound. To the best of our knowledge, the parameters of the asymptotically optimal codebooks are new.

In this paper, we give a construction of strongly regular Cayley graphs on the finite field F _qⁿ . As applications of these strongly regular Cayley graphs, a class of codebooks is presented and proved to be asymptotically optimal with respect to the Welch bound. Further more, these constructed codebooks have new parameters.