A generalized Kerdock code is a nonlinear (n, n2,[(q-1)/q](n-n))-code of length n=qm+1 over the field of q=2l elements (l≥1, m is odd). It is a concatenation of some special (base) linear code over the Galois ring of characteristic 4 and Reed-Solomon code of dimension 2. Here the complete weight enumerators of Kerdock code, base linear code and their analogues for even m are described. Incidentally, the weight characteristics of linear recurrences with the distinguished characteristic polynomial over the pointed Galois ring are indicated. Methods of proofs are based on the properties of trace function in Galois ring and quadrics over the field of characteristic 2.
Kuzmin, A., & Nechaev, A. (2001). Complete weight enumerators of generalized Kerdock code and related linear codes over Galois ring. Discrete Applied Mathematics, 111(1–2), 117–137. https://doi.org/10.1016/S0166-218X(00)00348-6