Does perfect secrecy imply uniform ciphertext distribution? I suspect the answer is no, but I am not able to either prove it, or provide an example. In Katz and Lindell's book, it is only said that with a perfectly secret encryption scheme, the plain and ciphertext distributions are independent. But when I try to construct an example with a non-uniform ciphertext distribution, (using say, 4 plaintexts as a message space), I cannot devise a plaintext and key distribution such that the resulting ciphertext distribution is not uniform.

What I am getting wrong? (or can anyone provide such an example?)
For a nonuniform construction with perfect secrecy, consider this scheme, with 2 bits of plaintext $(b_1, b_0)$, and four bits of key $(k_3, k_2, k_1, k_0)$.

The ciphertext consists of the three bits:
$$(k_3 \land k_2) \oplus b_0 \oplus k_0$$
$$b_1 \oplus k_1$$
$$b_0 \oplus k_0$$

This has perfect secrecy, in that for each ciphertexts, there is the same number of keys that map to any particular plaintext.  For example, for the ciphertext $(0,0,0)$, there are three keys that it to any plaintext, say, the plaintext $(1,0)$

This is nonuniform; as for random keys, the ciphertext $(0,0,0)$ will occur with probability three times that the ciphertext $(0,0,1)$ occurs at.