  A quasi-complementary sequence set (QCSS) refers to a set of two-dimensional
matrices with low non-trivial aperiodic auto- and cross- correlation sums. For
multicarrier code-division multiple-access applications, the availability of
large QCSSs with low correlation sums is desirable. The generalized Levenshtein
bound (GLB) is a lower bound on the maximum aperiodic correlation sum of QCSSs.
The bounding expression of GLB is a fractional quadratic function of a weight
vector $\mathbf{w}$ and is expressed in terms of three additional parameters
associated with QCSS: the set size $K$, the number of channels $M$, and the
sequence length $N$. It is known that a tighter GLB (compared to the Welch
bound) is possible only if the condition $M\geq2$ and $K\geq \overline{K}+1$,
where $\overline{K}$ is a certain function of $M$ and $N$, is satisfied. A
challenging research problem is to determine if there exists a weight vector
which gives rise to a tighter GLB for \textit{all} (not just \textit{some})
$K\geq \overline{K}+1$ and $M\geq2$, especially for large $N$, i.e., the
condition is {asymptotically} both necessary and sufficient. To achieve this,
we \textit{analytically} optimize the GLB which is (in general) non-convex as
the numerator term is an indefinite quadratic function of the weight vector.
Our key idea is to apply the frequency domain decomposition of the circulant
matrix (in the numerator term) to convert the non-convex problem into a convex
one. Following this optimization approach, we derive a new weight vector
meeting the aforementioned objective and prove that it is a local minimizer of
the GLB under certain conditions.
