The authors in this paper have proposed a multi-value sequence generated by a primitive polynomial, trace function, k-th power residue symbol, and a certain mapping function over the proper sub extension field. Here, the trace function actually maps an element of the extension field to an element of the proper sub extension field, which is actually a vector space. The distribution of numbers within the sequence becomes more balanced by considering the proper sub extension field. In addition, its period and autocorrelation properties also observed in this paper.