Disk Game

The game

Given parameters $(l,c,q)$. For convenience you can assume that $l$ is very large, $c=4$ and $q=1/2$. Consider the following game between a coder and a destroyer. The coder has spaces $f(\tau)$ indexed by $\tau\in 2^{\leq l}$ (all binary strings of length $\leq l$ which are called codes). He will allocate sources, which are also binary strings of length $\leq l$, into these $f(\tau)$. He will follow some requirements when doing the allocation.

• Each source $\sigma\in 2^{\leq l}$ is in $f(\tau)$ for some undestroyed $\tau$.
• For each $\sigma\in f(\tau),|\sigma|=|\tau|$.
• If $\tau$ is a prefix of $\tau’$, then each $\sigma’\in f(\tau’)$ has a prefix in $f(\tau)$.
• $|f(\tau)|\leq c$.

At the start of the game, the coder allocates sources until the requirements are met. Then in each round,

• The destroyer chooses a $\tau$ with $|\tau|=l$, and destroys that $\tau$.
• The coder allocates sources until the requirements are met.

The destroyer wins if the coder cannot meet the requirements before the end of the $(2^l\cdot q)^\text{th}$ round. The coder wins if he survives this number of rounds.

The problem

Question. Is there $c$ and $q$ such that the coder wins in the game $(l,c,q)$ for all $l$? What is the winning strategy?