In this paper, a new family of binary sequences of period 2~n - 1 with low correlation is proposed for integer n = em and even m. The new family has family size 2" + 1 and maximum nontrivial correlation 2~((n+e)/2) and 2~((n+e-1)/2) + 1 for even and odd e respectively. Especially, for n = 2m and 3m, we obtain a new family of binary sequences with maximum nontrivial correlation 2~(n/2) + 1, and the obtained family is one of the binary families with best correlation among the known families with family size no less than their period 2~n - 1 for even n. Moreover, the correlation distribution of the new family is also determined.
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