Cryptographic boolean functions via group Hadamard matrices

For any integers n,m, 2n > m > n we construct a set of boolean functions on Vm, say {f1(z),...,fn(z)}, which has the following important cryptographic properties: (i) any nonzero linear combination of the functions is balanced; (ii) the nonlinearity of any nonzero linear combination of the functions is at least 2m-1 - 2n-1; (iii) any nonzero linear combination of the functions satisfies the strict avalanche criterion; (iv) the algebraic degree of any nonzero linear combination of the functions is m - n + 1; (v) F(z) = (f1(z),...,fn(z))runs through each vector in Vn precisely 2m-n times while z runs through Vm.