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 F
_q
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.