Extremal character spectra of finite projective systems
Abstract
Let ℳ be a finite spanning multiset of points in PG(k−1,q), and let Charℳ be its character function. For 1≤ r≤ k, we study the maximum number of distinct values attained by Charℳ 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≤ r<k, put Mr=qbinomkr and Br=qbinomk−1r. The minimum size νr(k,q) of an r-MCS system satisfies ⌈binomMr2/Br⌉≤ νr(k,q)≤ binomMr−12+(qk−qk−r)/(q−1). The upper bound is at most binomMr2, strictly less when r≤ 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.
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BibTeX
@misc{li2026extremal,
title = {{Extremal character spectra of finite projective systems}},
author = {Shitao Li and Yang Li and Gaojun Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.42006},
eprint = {2610.42006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.42006},
}
RIS
TY - UNPB
TI - Extremal character spectra of finite projective systems
AU - Shitao Li
AU - Yang Li
AU - Gaojun Luo
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.42006
UR - https://arxiv.org/abs/2610.42006
AB - Let ℳ be a finite spanning multiset of points in PG(k−1,q), and let Char_ℳ be its character function. For 1≤ r≤ k, we study the maximum number of distinct values attained by Char_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≤ r<k, put M_r=qbinomkr and B_r=qbinomk−1r. The minimum size ν_r(k,q) of an r-MCS system satisfies ⌈binomM_r2/B_r⌉≤ ν_r(k,q)≤ binomM_r−12+(q^k−q^k−r)/(q−1). The upper bound is at most binomM_r2, strictly less when r≤ 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.
ER -