For odd n=2l+1 and an integer /spl rho/ with 1/spl les//spl rho//spl les/l, a new family S/sub o/(/spl rho/) of binary sequences of period 2/sup n/-1 is constructed. For a given /spl rho/, S/sub o/(/spl rho/) has maximum correlation 1+2/sup n+2/spl rho/-1/2/, family size 2/sup n/spl rho//, and maximum linear span n(n+1)/2. Similarly, a new family of S/sub e/(/spl rho/) of binary sequences of period 2/sup n/-1 is also presented for even n=2l and an integer /spl rho/ with 1/spl les//spl rho/<l, where maximum correlation, family size, and maximum linear span are 1+2/sup n/2+/spl rho//,2/sup n/spl rho//, and n(n+1)/2, respectively. The new family S/sub o/(/spl rho/) (or S/sub e/(/spl rho/)) contains Boztas and Kumar's construction (or Udaya's) as a subset if m-sequences are excluded from both constructions. As a good candidate with low correlation and large family size, the family S/sub o/(2) is discussed in detail by analyzing its distribution of correlation values.