What are the equations (Gen, Enc, Dec) for the vigenere cipher?

Long story short, trying to understand how I would write the vigenere cipher mathematically when using Gen, Enc, Dec and I can't figure it out. This is what I've come up with so far.

$$\mathrm{Gen}: k ={0…25}^t$$

$$\mathrm{Enc}: c_i = (p_i + k_i) \pmod {26}$$

$$\mathrm{Dec}: p_i = (c_i – k_i) \pmod {26}$$

It doesn't really seem right though, so that's why I'm asking.

The only way I'd change it is to make it explicit that $k$ is sampled from that set rather than is equal to it and to note that the key is used cyclically: \begin{eqnarray*} \mathrm{Gen}:&&k\leftarrow \{0,\ldots,25\}^t\\ \mathrm{Enc}:&&c_i=p_i+k_{i\pmod t}\pmod{26}\\ \mathrm{Dec}:&&p_i=c_i-k_{i\pmod t}\pmod{26} \end{eqnarray*}