I'm studying for exam that is coming up and I'm reading about correcting errors in linear code and I'm struggling with this problem. Any solutions? Thanks!
Suppose we wish to be able to send $256$ different messages by means of a linear code which will correct one error. (a) What is the least possible length of such a code? (b) Find a lower bound for the length of the code if two errors are to be corrected.
Welcome to MSE! I'd consider the Hamming bound, $$q^n \geq |C|\sum_{i=0}^t {n\choose i}(q-1)^i,$$ where $|C|=2^8$, $q=2$, $d=2t+1=3$, i.e., $t=1$.
Search for the smallest block length $n\geq 8$ such that this inequality is satisfied.