I am not sure I interpret what is meant by “cross-correlation $2^{a−1}b$” correctly. Let $a=6,b=1,k=1$. I have $w_0 = (01)^{16}.$ Let $a=6,b=1,k=2$. I have $w_1 = (0011)^{8}.$ Do these words satisfy the answer? Then $2^{6−1} \times 1 = 32,$ which is equal to the size of the words. But $32$ is too large for the value of cross-correlation between $w_0$ and $w_1.$

@PeterTaylor: “If you remove the first condition, for any even n you can get a 1-element set T by taking an arbitrary balanced 2n-bit word” — but $f$ requires exactly two elements to operate, so a 1-element set cannot solve this particular problem. It is undefined for a single argument. “the question becomes what the largest T is which respects properties 2 and 3” — I think that at least a set with the smallest possible cardinality (i.e. $2$) will be enough for me, so I have edited the question accordingly.