arXiv subjects

Ferdinand Ihringer

Publications and source records attributed to Ferdinand Ihringer.

At least 37 records · Page 2Linked to original sources

arXiv2024

A common generalization of hypercube partitions and ovoids in polar spaces

We investigate what we call generalized ovoids, that is families of totally isotropic subspaces of finite classical polar spaces such that each maximal totally isotropic subspace contains precisely one member of that family. This is a generalization of ovoids in polar spaces as well as the natural $q$-analog of a subcube partition of the hypercube (which can be seen as a polar space with $q=1$). Our main result proves that a generalized ovoid of $k$-spaces in polar spaces of large rank does not exist. More precisely, for $q=p^h$, $p$ prime, and some positive integer $k$, a generalized ovoid of $k$-spaces in a polar space $\mathcal{P}$ with rank $r \geq r_0(k, p)$ in a vector space $V(n,q)$ does not exist.

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arXiv2024

The strongly regular twisted $D_{5,5}(q)$ graph

We construct a new family of strongly regular graphs with the same parameters as the strongly regular graphs $D_{5,5}(q)$. The construction can be seen as a variant of the construction of twisted Grassmann graphs by Van Dam and Koolen.

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arXiv2023

On MSR Subspace Families of Lines

A minimum storage regenerating (MSR) subspace family of $\mathbb{F}_q^{2m}$ is a set $\mathcal{S}$ of $m$-spaces in $\mathbb{F}_q^{2m}$ such that for any $m$-space $S$ in $\mathcal{S}$ there exists an element in $\mathrm{PGL}(2m, q)$ which maps $S$ to a complement and fixes $\mathcal{S} \setminus \{ S \}$ pointwise. We show that an MSR subspace family of $2$-spaces in $\mathbb{F}_q^4$ has at most size $6$ with equality if and only if it is a particular subset of a Segre variety. This implies that an $(n, n-2, 4)$-MSR code has $n \leq 9$.

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arXiv2023

Local recognition of the point graphs of some Lie incidence geometries

Given a finite Lie incidence geometry which is either a polar space of rank at least $3$ or a strong parapolar space of symplectic rank at least $4$ and diameter at most $4$, or the parapolar space arising from the line Grassmannian of a projective space of dimension at least $4$, we show that its point graph is determined by its local structure. This follows from a more general result which classifies graphs whose local structure can vary over all local structures of the point graphs of the aforementioned geometries. In particular, this characterises the strongly regular graphs arising from the line Grassmannian of a finite projective space, from the half spin geometry related to the quadric $Q^+(10,q)$ and from the exceptional group of type $\mathsf{E_6}(q)$ by their local structure.

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arXiv2023

Affine vector space partitions

An affine vector space partition of $\operatorname{AG}(n,q)$ is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.

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arXiv2023

The proportion of non-degenerate complementary subspaces in classical spaces

Given positive integers $e_1,e_2$, let $X_i$ denote the set of $e_i$-dimensional subspaces of a fixed finite vector space $V=(\mathbb{F}_q)^{e_1+e_2}$. Let $Y_i$ be a non-empty subset of $X_i$ and let $α_i=|Y_i|/|X_i|$. We give a positive lower bound, depending only on $α_1,α_2,e_1,e_2,q$, for the proportion of pairs $(S_1,S_2)\in Y_1\times Y_2$ which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.

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arXiv2022

Degree 2 Boolean Functions on Grassmann Graphs

We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are known for $n \geq 5$. This paper focusses on providing a list of examples on the case $d=2$ in general dimension and in particular for $(n, k)=(6,3)$ and $(n,k) = (8, 4)$. We also discuss connections to the analysis of Boolean functions, regular sets/equitable bipartitions/perfect 2-colorings in graphs, $q$-analogs of designs, and permutation groups. In particular, this represents a natural generalization of Cameron-Liebler line classes.

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arXiv2022

Large $(k; r, s; n, q)$-sets in Projective Spaces

A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q \rightarrow \infty$. In particular, we show the existence of $(3, 2)$-sets of size $(1+o(1)) q^{3/2}$ for $n=6$, $(4, 2)$-sets of size $(1+o(1)) q^{\frac{n-1}{2}}$, and $(9, 2)$-sets of size $(1+o(1)) q^2$ for $n=4$. We also generalize a bound by Rao from 1947 and show that an $(r,s)$-set has size at most $O(q^{\frac{n-e+1}{e}})$ if there exist integers $d,e \geq 2$ such that $s=d(e-1)$ and $r=de-1$.

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arXiv2022

Approximately Strongly Regular Graphs

We give variants of the Krein bound and the absolute bound for graphs with a spectrum similar to that of a strongly regular graph. In particular, we investigate what we call approximately strongly regular graphs. We apply our results to extremal problems. Among other things, we show the following: (1) Caps in $\mathrm{PG}(n, q)$ for which the number of secants on exterior points does not vary too much, have size at most $O(q^{\frac34 n})$ (as $q \rightarrow \infty$ or as $n \rightarrow \infty$). (2) Optimally pseudorandom $K_m$-free graphs of order $v$ and degree $k$ for which the induced subgraph on the common neighborhood of a clique of size $i \leq m-3$ is similar to a strongly regular graph, have $k = O(v^{1 - \frac{1}{3m-2i-5}})$.

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arXiv2022

Switching for Small Strongly Regular Graphs

We provide an abundance of strongly regular graphs (SRGs) for certain parameters $(n, k, λ, μ)$ with $n < 100$. For this we use Godsil-McKay (GM) switching with a partition of type $4,n-4$ and Wang-Qiu-Hu (WQH) switching with a partition of type $3,3,n-6$ or $4,4,n-8$. In most cases, we start with a highly symmetric graph which belongs to a finite geometry. Many of the obtained graphs are new; for instance, we find 16565438 strongly regular graphs with parameters $(81, 30, 9, 12)$ while only 15 seem to be described in the literature. We provide statistics about the size of the occurring automorphism groups. We also find the recently discovered Krčadinac partial geometry, thus finding a third method of constructing it.

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arXiv2022

Non-Geometric Cospectral Mates of Line Graphs with a Linear Representation

For an incidence geometry $\mathcal{G} = (\mathcal{P}, \mathcal{L}, \text{I})$ with a linear representation $\mathcal{T}_n^*(\mathcal{K})$, we apply WQH switching to construct a non-geometric graph $Γ'$ cospectral with the line graph $Γ$ of $\mathcal{G}$. As an application, we show that for $h \geq 2$ and $0 < m < h$, there are strongly regular graphs with parameters $(v, k, λ, μ) = (2^{2h} (2^{m+h}+2^m-2^h), 2^h (2^h+1)(2^m-1), 2^h (2^{m+1}-3), 2^h (2^m-1))$ which are not point graphs of partial geometries of order $(s,t,α) = ((2^h+1)(2^m-1), 2^h-1, 2^m-1)$.

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arXiv2022

The least Euclidean distortion constant of a distance-regular graph

In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin's conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph of diameter $d$, the most contracted pairs of vertices are those at distance $d$. In this paper, we confirm Vallentin's conjecture for several families of distance-regular graphs. We also provide counterexamples to this conjecture, where the largest contraction occurs between pairs of vertices at distance $d{-}1$. We suggest three alternative conjectures and prove them for several families of distance-regular graphs.

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arXiv2021

Graphs cospectral with NU$(n + 1, q^2)$, $n \ne 3$

Let $H(n, q^2)$ be a non-degenerate Hermitian variety of $PG(n, q^2)$, $n \ge 2$. Let NU$(n+1, q^2)$ be the graph whose vertices are the points of $PG(n, q^2) \setminus H(n, q^2)$ and two vertices $P_1$, $P_2$ are adjacent if the line joining $P_1$ and $P_2$ is tangent to $H(n, q^2)$. Then NU$(n + 1, q^2)$ is a strongly regular graph. In this paper we show that NU$(n + 1, q^2)$, $n \ne 3$, is not determined by its spectrum.

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arXiv2021

Cameron-Liebler $k$-sets in $\text{AG}(n,q)$

We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\text{AG}(n, q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\text{PG}(n, q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\text{AG}(n, q)$ correspond to certain equitable bipartitions of the Association scheme of $k$-spaces in $\text{AG}(n, q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\text{AG}(n, q)$. We define Cameron-Liebler $k$-sets in $\text{AG}(n, q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\text{AG}(n, q)$ and $\text{PG}(n, q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets. This paper focuses on $\text{AG}(n, q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\text{AG}(3, q)$ was already treated separately.

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arXiv2021

Ovoids of Generalized Quadrangles of Order $(q, q^2-q)$ and Delsarte Cocliques in Related Strongly Regular Graphs

We investigate strongly regular graphs for which Hoffman's ratio bound and Cvetcović's inertia bound are equal. This means that $ve^- = m^-(e^- - k)$, where $v$ is the number of vertices, $k$ is the regularity, $e^-$ is the smallest eigenvalue, and $m^-$ is the multiplicity of $e^-$. We show that Delsarte cocliques do not exist for all Taylor's $2$-graphs and for point graphs of generalized quadrangles of order $(q,q^2-q)$ for infinitely many $q$. For cases where equality may hold, we show that for nearly all parameter sets, there are at most two Delsarte cocliques.

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arXiv2020

A Note on a Construction by Pavese and Smaldore

Recently, Pavese and Smaldore constructed graphs cospectral to $NU(5, q^2)$ for $q>2$. We show that their construction works for $NU(n, q^2)$, $n \geq 5$. Hence, none of the graphs $NU(n, q^2)$, $n \geq 5$, are determined by their spectrum.

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arXiv2020

New and Updated Semidefinite Programming Bounds for Subspace Codes

We show that $A_2(7,4) \leq 388$ and, more generally, $A_q(7,4) \leq (q^2-q+1)[7]_q + q^4 - 2q^3 + 3q^2 - 4q + 4$ by semidefinite programming for $q \leq 101$. Furthermore, we extend results by Bachoc et al. on SDP bounds for $A_2(n,d)$, where $d$ is odd and $n$ is small, to $A_q(n,d)$ for small $q$ and small $n$.

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arXiv2020

Boolean degree 1 functions on some classical association schemes

We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as \textit{completely regular strength 0 codes of covering radius 1}, \textit{Cameron--Liebler line classes}, and \textit{tight sets}. We classify all Boolean degree $1$ functions on the multislice. On the Grassmann scheme $J_q(n, k)$ we show that all Boolean degree $1$ functions are trivial for $n \geq 5$, $k, n-k \geq 2$ and $q \in \{ 2, 3, 4, 5 \}$, and that for general $q$, the problem can be reduced to classifying all Boolean degree $1$ functions on $J_q(n, 2)$. We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree $1$ functions are trivial for appropriate choices of the parameters.

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