Trace-function on a Galois ring in coding theory

Patterns of the distribution of elements in words of linear codes over a Galois ring and in their representations over a Galois field are investigated. Often they may be evaluated using numbers of some special solutions of the equation defined by the trace-function on a Galois ring. Here the solutions of such an equation over a Galois ring R=GR(q2,4) of characteristic 4 are enumerated. It allows us in particular to describe the complete weight enumerators of the base linear code K R (m) and the appropriate Kerdock code K q (m+1) over a Galois field of q=21 elements.Results based on properties of special quadrics over GF(21) arise by description of the 2-adic decomposition of the trace-function.