# Proof of the minimum distance of Reed-Solomon codes

How we can prove that the t-error-correcting Reed-Solomon code with symbols from $$GF(2^m)$$ generated by $$g(X)=(X+\alpha)(X+\alpha^2)...(X+\alpha^{2t})$$ has minimum distance exactly $$2t+1$$ where $$\alpha$$ is a primitive element in $$GF(2^m)$$.
I remember one theorem that says no $$2t$$ columns of its parity-check matrix must not add to zero, but have no idea to use this.

A $$t$$-error-correcting code must have minimum distance at least $$2t+1$$, not $$2t-1$$ as you claim. Now, the Singleton bound tells us that a $$[n,k]$$ code has minimum distance at most $$n-k+1$$, but since $$n-k$$ equals the degree of the generator polynomial (the degree is $$2t$$), we have that $$d \leq 2t+1.$$ On the other hand, the BCH bound tells us that if $$2t$$ consecutive powers of $$\alpha$$ are roots of the generator polynomial, then the minimum distance of the code is at least $$2t+1$$, that is, $$d \geq 2t+1$$. Comparing these two bounds, we arrive at the conclusion that $$d = 2t+1$$ exactly without needing to look at paritycheck matrices and the like.