Determining When the Absolute State Complexity of a Hermitian Code Achieves Its DLP Bound

Let g be the genus of the Hermitian function field $H/{\mathbb F}_{q^2}$ and let $C_{\cal L}(D,mQ_{\infty})$ be a typical Hermitian code of length n. In [Des. Codes Cryptogr., to appear], we determined the dimension/length profile (DLP) lower bound on the state complexity of $C_{\cal L}(D,mQ_{\infty})$. Here we determine when this lower bound is tight and when it is not. For $m\leq \frac{n-2}{2}$ or $m\geq \frac{n-2}{2}+2g$, the DLP lower bounds reach Wolf's upper bound on state complexity and thus are trivially tight. We begin by showing that for about half of the remaining values of m the DLP bounds cannot be tight. In these cases, we give a lower bound on the absolute state complexity of $C_{\cal L}(D,mQ_{\infty})$, which improves the DLP lower bound. Next we give a "good" coordinate order for $C_{\cal L}(D,mQ_{\infty})$. With this good order, the state complexity of $C_{\cal L}(D,mQ_{\infty})$ achieves its DLP bound (whenever this is possible). This coordinate order also provides an upper bound on the absolute state complexity of $C_{\cal L}(D,mQ_{\infty})$ (for those values of $m$ for which the DLP bounds cannot be tight). Our bounds on absolute state complexity do not meet for some of these values of m, and this leaves open the question whether our coordinate order is best possible in these cases. A straightforward application of these results is that if $C_{\cal L}(D,mQ_{\infty})$ is self-dual, then its state complexity (with respect to the lexicographic coordinate order) achieves its DLP bound of $\frac {n}{2}-\frac{q^2}{4}$, and, in particular, so does its absolute state complexity.