Designs in additive codes over GF(4) (Q1412248)

Additive codes over GF(4) are of interest because of their connection to additive codes. Höhn showed that certain vectors in some additive self-dual codes over GF(4) hold generalized \(t\)-designs. There are two types of designs. One is a classical \(t\)-design with repeated blocks. In this case there is an analog of the Assmus-Mattson theorem for additive codes over GF(4). The other is a generalized \(t\)-design first introduced by Delsarte. As an example, the unique additive self-dual (\(12,12^{12},6\)) code (dodecacode) is considered. It is shown that there exists a 5-\((12,6,3)\) design in this code with either 3 distinct blocks or three repeated blocks covering a 5-set. In addition, new simple 3-\((11,5,4)\) designs are obtained from the shortened dodecacode. Any extremal additive even self-dual code is shown to be homogeneous.