  In this paper we study a family of binary Legendre sequences and its family
complexity. Family complexity is a pseudorandomness measure introduced by
Ahlswede et.~al.~in 2003. A lower bound on the family complexity of a family
based on the Legendre symbol of polynomials over a finite field was given by
Gyarmati in 2015. In this article we improve the bound given by Gyarmati on
family complexity of binary Legendre sequences. The bound depends on the
Lambert W function and the number of elements in a finite field belonging to
its proper subfield. Moreover, we present a fast method for calculating the
bound.
