The distribution of bit patterns is an important measure to check the randomness of a sequence. The authors of this paper observed this crucial property in a binary sequence which generated by using a primitive polynomial, trace function, and Legendre symbol defined over the sub extension field. The authors create a new dimension in the sequence generation research area by considering the sub extension field, whereas all our previous works are focused in the prime field. In terms of the distribution of bit patterns property, this research work has notable outcomes more specifically the binary sequence (defined over the sub extension field) holds much better (close to uniform) bit distribution than the previous binary sequence (defined over the prime field). Furthermore, the authors theoretically proved the distribution of bit property in this paper.