Extremal character spectra of finite projective systems

Abstract

Let $\mathcal M$ be a finite spanning multiset of points in ${\rm PG}(k-1,q)$, and let ${\rm Char}_{\mathcal M}$ be its character function. For $1\le r\le k$, we study the maximum number of distinct values attained by ${\rm Char}_{\mathcal M}$ on the codimension-$r$ flats under a total-multiplicity budget. The spectrum is subject to two natural upper bounds, whose extremal cases lead respectively to $r$-maximum character spectrum ($r$-MCS) and $r$-full character spectrum ($r$-FCS) systems. We establish the existence of $r$-FCS systems in a broad parameter range, and solve the size-constrained extremal problem completely when $r=k-1$, including the minimum-cost multiplicity profiles. For $1\le r<k$, put $M_r=\qbinom{k}{r}$ and $B_r=\qbinom{k-1}{r}$. The minimum size $ν_r(k,q)$ of an $r$-MCS system satisfies \[\left\lceil\frac{\binom{M_r}{2}}{B_r}\right\rceil\le ν_r(k,q)\le \binom{M_r-1}{2}+\frac{q^k-q^{k-r}}{q-1}.\] The upper bound is at most $\binom{M_r}{2}$, strictly less when $r\le k-2$, and exact when $r=k-1$. Its proof applies positive additive labelings to the complement of one flat, whose trace is empty while all remaining traces are distinct. For $r=1$, we improve the known upper bounds on the minimum length of such systems. We also develop global moment identities and local flat-incidence bounds that impose strong structural constraints on large character spectra. Under the projective-system correspondence, complementary character spectra are the support-weight spectra of $r$-dimensional subcodes.