Distances in the collinearity graphs of geometries related to simplex codes
Abstract
Let cS(k,q) be the point-line geometry whose points and lines correspond to one-dimensional and two-dimensional subcodes of q-ary simplex codes of dimension k, respectively. We determine the complete distance relation and the diameter of its collinearity graph. For two points whose representing vectors have different sets of zero coordinates there is a common neighbour. If the representing vectors have the same zero set, then a common neighbour exists precisely when (q−1)qk−2 of their non-zero coordinate ratios can be selected so that their product is one. To prove this, we use a generalization of Hall's result for finite abelian groups. Applying the Erdős--Ginzburg--Ziv theorem, we establish that the above product condition holds for k≥3, which implies that the diameter is two. If k=2, then the diameter is one when q=2, two when q=3 and three when q≥4.
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BibTeX
@misc{tyc2026distances,
title = {{Distances in the collinearity graphs of geometries related to simplex codes}},
author = {Adam Tyc},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00916},
eprint = {2611.00916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00916},
}
RIS
TY - UNPB
TI - Distances in the collinearity graphs of geometries related to simplex codes
AU - Adam Tyc
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00916
UR - https://arxiv.org/abs/2611.00916
AB - Let cS(k,q) be the point-line geometry whose points and lines correspond to one-dimensional and two-dimensional subcodes of q-ary simplex codes of dimension k, respectively. We determine the complete distance relation and the diameter of its collinearity graph. For two points whose representing vectors have different sets of zero coordinates there is a common neighbour. If the representing vectors have the same zero set, then a common neighbour exists precisely when (q−1)q^k−2 of their non-zero coordinate ratios can be selected so that their product is one. To prove this, we use a generalization of Hall's result for finite abelian groups. Applying the Erdős--Ginzburg--Ziv theorem, we establish that the above product condition holds for k≥3, which implies that the diameter is two. If k=2, then the diameter is one when q=2, two when q=3 and three when q≥4.
ER -