For odd and an integer , a new family of binary sequences with period is constructed. For a given , has maximum correlation "! # $ %'& ( $ , family size ) , and maximum linear span *+ ,.-0/ 1 . Similarly, a new family of .2 of binary sequences with period is also presented for even 3 4 and an integer 5 76 , where maximum correlation, family size, and maximum linear span are 3 ! $ ,8) , ) , *9 ,.-:/ 1 , respectively. The new family (or 2 ) contains Boztas and Kumar’s construction  (or Udaya’s ) as a subset if ; -sequences are excluded from both constructions. As a good candidate with both low correlation and large family size, the family is discussed in detail by analyzing its distribution of correlation values.