I am looking at the proof of the singleton bound and I don't understand the first step. I want to show that $A_q(n,d)\leq q^{n-d+1}$ where $A_q(n,d)$ is the code of maximal size given these parameters, d is the distance of the code, n the length, q the size of the alphabet.
The first step consist in the erasing of the first $d-1$ coordinates of an arbitrary code of distance $d$. After that the claim is that the resulting code has the same size of the original one since the distance is d. But take for instance the following binary code
$(110),(101),(011),(111)$
Then the code has distance 2 and erasing the first coordinate I get
$(10),(01),(11),(11)$
The new code has size 3 not 4 as the original one. Here the link to the statement and the proof: http://en.m.wikipedia.org/wiki/Singleton_bound
What is my mistake? Thanks
That code actually has distance 1, since $d(110, 111) = 1$.