I am trying to self-educate about m-sequences, which led me to the topic of the Welch lower bounds on the maximum cross-correlation of sets of vectors in $\mathbb{C}^n$. The Wikipedia page "Welch bounds", following the 1974 paper by LR Welch, starts off as follows:
"If $\{x_1,\ldots,x_m\}$ are unit vectors in $\mathbb{C}^n$, define $c_\max = \max_{i\neq j} |\langle x_i, x_j \rangle|$, where $\langle\cdot,\cdot\rangle$ is the usual inner product on $\mathbb{C}^n$". There is NO stipulation that the $m$ chosen vectors are equiangular or have any other special relationship. It is tacitly assumed that all possible pairs in the arbitrarily chosen set of m vectors should be compared by evaluating their inner products. A formula is then derived for $c_\max$ as a function of $m$, $n$, and a dimensional parameter $k$.
I feel that something has been left out of the initial statement. Although the statement above forbids correlating any vector with itself ($c_{ii} = 1$), no other restriction is placed on our choice of vectors. Therefore we can always choose 2 unit vectors that almost coincide (separated by angle $\theta$), and as $\theta \rightarrow 0, |\langle x_i, x_j \rangle| \rightarrow 1$. Since $c_\max$ is defined as the maximum value over the chosen set of $m$ vectors, and we appear to be allowed free choice in selecting any $m$ vectors from the infinity of unit vectors on $\mathbb{C}^n$, then we can always chose m vectors so that they include a pair of almost coincident vectors with $c_{i,j} = 1 - \epsilon$, where $\epsilon$ is as small as we please. Therefore $c_\max$ must always be $1$, and this does not depend on $m$, $n$, or $k$.
Obviously, this interpretation contradicts the whole point of the bounds derivation. So what have they omitted in their statement of the problem, or what have I misunderstood?
The Welch bound is a bound from below. Hence, the interesting part is not how large you can make $c_{\mathrm{max}}$ but how small you can make it.
Geoetrically: The question "How "orthogonal" can you expect many vectors to be? " is the same as asking how small the Welch bound is.