The step-count busy beaver function relative to the chosen machine coding

Fix the chosen machine code $<mi\; mathvariant="normal">⌜</mi>M<mi\; mathvariant="normal">⌝</mi>$ from A fixed effective binary encoding of deterministic one-tape Turing machines. For $n\in \mathbb{N}$, let $BB(n)$ be the maximum number of steps taken by any coded deterministic one-tape Turing machine $M$ such that:

• the code length satisfies $\mathrm{\mid }<mi\; mathvariant="normal">⌜</mi>M<mi\; mathvariant="normal">⌝</mi>\mathrm{\mid }\le n$, and
• $M$ halts on the empty input word $\epsilon$.

If no such machine exists, define $BB(n):=0$.

Thus $BB(n):=\max \{{\mathrm{time}}_M(\epsilon ):\mathrm{\mid }<mi\; mathvariant="normal">⌜</mi>M<mi\; mathvariant="normal">⌝</mi>\mathrm{\mid }\le n,M\text{ halts on }\epsilon \},$ with the empty-set convention above.

• This is a coding-dependent busy beaver function: changing the chosen machine syntax changes the finite search space for each $n$.

• The maximization is over halting time on the blank input, not over number of nonblank symbols written.