Designed distance ($d(C)\ge \delta$)

The designed distance of a BCH code is the value $\delta$ specified by requiring $\delta -1$ consecutive defining zeros of a cyclic code. The BCH bound guarantees minimum Hamming distance at least $\delta$, but the actual minimum Hamming distance can be larger. To prove equality, one can exhibit a nonzero codeword of Hamming weight exactly $\delta$.