According to this paper in page $4$ where it describes the encryption scheme where a cipher function is defined as it follows $$\rho:T\times Y \to X$$ such that $|Y|\geq |T|$, where $y\in Y$ is the key and $x\in X$ a code. $t$, $x$ ,$y$ are variables that are defined in the corresponding fields $T$, $X$, $Y$ and $\rho(\cdot,y)$ is a bijection namely every pair of $(x,y)$ is associated with only one $t$ (i.e. $\rho(t,y)=x$).
Could we re-define the $\rho$ function in the following way and still be bijective? That is
$$\rho:T\times(\Pi_{i=1}^K Y_i)\to X$$ where $(y_1,y_2.\cdots,y_K)$ is family of i.i.d. uniform random variables such that the $(K+1)$- tuple $(x,y_1,y_2,\cdots, y_K)$, is associated with only one $t$.