Let's assume I have $N$ binary strings $\{T_1,T_2,\ldots,T_N\}$ of length $L$. All these strings satisfy a minimum hamming distance with respect to a reference binary string R with $\|R\|_1$ ones and the same length of $L$, i.e.,
$$d(T_i,R)\leq \left\lfloor{\frac{L-\|R\|_1}{2}}\right\rfloor+1$$
The Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. For example the hamminf distance of 001 and $111$ is $2$.
What is the probability that for a single bit, e.g. the $K$th bit,
$$|N\times R[k]-\sum\limits_{i=1}^N T_i[k]|\geq\left\lfloor\frac{N}{2}\right\rfloor$$
Note: In fact we have a conditional probability conditioned on the samples satisfying the preceeding hamming distance condition)