Is there a lower bound of number of redundant bits necessary to encode a word with certain Hamming distance?
As Tsuyoshi points out in the comments, there are a number of such bounds. However, for the sake of actually giving you an answer, let me point you to the Singleton bound, which states that for a $(s,N,d)$-code over $\mathbb{F}_b$ that $N\leq b^{s-d+1}$.