On mixing linear transforms for block ciphers

We consider AES-type block ciphers over the finite field with linear mixing transforms of three classes: maximally mixing, block-uniform and composite. Transition probability matrices of pairs of input blocks for these schemes with random indepеndent equiprobable round keys are investigated. A minimum number of rounds sufficient for the generation of doubly transitive set of permutations are found for schemes without mixing transforms in odd rounds.