# Maximum characters in a deterministic Turing machine

Assume we have a deterministic Turing machine $$M = (q_s, q_a, q_r, \Sigma, \Gamma, \delta, Q, b)$$ where $$q_s,q_a,q_r$$ are the (unique) starting state, accept state and reject state respectively, $$Q$$ the set of non-final states, $$\Sigma$$ the input alphabet, $$\Gamma$$ the tape alphabet, $$\delta$$ the transition function and $$b \in \Gamma$$ the blank symbol.

How many characters can fit in $$\Gamma$$, as a function of $$|\Sigma|, |Q|$$ such that for each $$c \in \Gamma$$, $$\delta$$ will be defined by it for some state and character?