What characteristics define a function to be a block cipher
The Wikipedia definition states;


  In cryptography, a block cipher is a deterministic algorithm operating on fixed-length groups of bits, called a block, with an unvarying transformation that is specified by a symmetric key





  A block cipher consists of two paired algorithms, one for encryption, E, and the other for decryption, D. Both algorithms accept two inputs: an input block of size n bits and a key of size k bits; and both yield an n-bit output block. The decryption algorithm D is defined to be the inverse function of encryption.


Your $F_k^r(m) := r(k,m)$ with $r: \{0,1\}^{3n} \rightarrow \{0,1\}^n$ is not a block cipher since:


The input block size and the output block size are not same as in the Wikipedia definition. Therefore you cannot define the inverse (decryption). With the key concerned, the input space must be $2^{2n}$, not $2^{3n}$ and output space must be $2^{n}$.