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In this paper, Zopf4-valued quadratic forms defined on a vector space over GF(2) are studied. A classification of such forms is established, distinguishing Zopf4-valued quadratic forms only by their rank and whether the associated bilinear form is alternating. This result is used to compute the distribution of certain exponential sums, which occur frequently in the analysis of quaternary codes and quaternary sequence sets. The concept is applied as follows. When t=0 or m is odd, the correlation distribution of family S(t), consisting of quaternary sequences of length 2 m-1, is established. Then, motivated by practical considerations, a subset S *(t) of family S(t) is defined, and the correlation distribution of family S *(t) is given for odd and even m.