In this paper,we calculate the number of the codewords(with Hamming weight 7)of each type in the Preparata codes over Z4,then give the parameter sets of 3-designs constructed from the supports of the codewords of each type.Moreover,we prove that the first two families of 3-designs are simple and the third family of the 3-designs has repeated blocks.
In this paper, a bivariate generating function CF(x, y) =f(x)-yf(xy)1-yis investigated, where f(x)= n 0fnxnis a generating function satisfying the functional equation f(x) = 1 + r j=1 m i=j-1aij xif(x)j.In particular, we study lattice paths in which their end points are on the line y = 1. Rooted lattice paths are defined. It is proved that the function CF(x, y) is a generating function defined on some rooted lattice paths with end point on y = 1. So, by a simple and unified method, from the view of lattice paths, we obtain two combinatorial interpretations of this bivariate function and derive two uniform partitions on these rooted lattice paths.
