- the trace representation and the 2 - adic representation for the dual of the quaternary goethals code are given , and its binary image is proved to be the formal dual of the binary goethals code . - the generalized macwilliams identities are given for the support weight distribution of a quaternary linear code with type 4k and that of its dual code 给出了四元goethals码的对偶码的迹表示及2 - adic表示，并证明了它在gray映射下的二元像是二元goethals码的形式对偶码；从而可以将对非线性码二元goethals码及其形式对偶码的研究转化为对四元线性码的研究； 3
