arXiv subjects

Ming-Hsuan Kang

Publications and source records attributed to Ming-Hsuan Kang.

At least 19 recordsLinked to original sources

arXiv2026

Character sums on an oriented singer conic and explicit Ramanujan double covers

Let $q$ be odd. The trace conic in $\mathbb{F}_{q^3}$ determines a Singer difference set in $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^\times$ and a natural square-class lift to $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^{\times 2}$. We study the odd multiplicative Fourier coefficients of this lift and prove that they are bounded in absolute value by $2\sqrt{q}$. The proof improves the naive six-puncture Weil bound by exploiting a projective Klein-four symmetry of the associated rank-one local system. The resulting nontrivial cocycle produces a quaternionic action on its four-dimensional cohomology, while Frobenius symmetry reduces the relevant trace to two Weil-scale eigenvalues. As an application, the oriented conic yields an explicit Singer-invariant signing of the point-line incidence graph of $\mathrm{PG}(2, q)$. The corresponding dihedral Cayley graph is a connected Ramanujan double cover. Thus a conic lift already known in finite-geometric constructions has an additional Ramanujan spectral property governed by its odd multiplicative character sums.

math.NT↗

arXiv2026

Linear programming bounds for binary and ternary LCD Codes

We derive linear programming (LP) bounds on the minimum distance of binary and ternary linear complementary dual (LCD) codes by imposing arithmetic constraints on their weight enumerators. Special values of the weight enumerator give finitely many Gauss phases, each of which yields linear equations in the ordinary weight-distribution variables. The resulting bounds strengthen the real-valued LCD counting LP without introducing additional variables; both the number of branches and the number of added equations per branch are bounded independently of the code length. Exact certificates establish strict improvements for 62 binary parameter pairs of length at most 20 and 39 ternary pairs of length at most 14. For four binary pairs, the bounds also improve the joint-weight-enumerator LP while using fewer variables per branch. These comparisons show that Gauss-phase information provides a compact strengthening of existing LP relaxations for LCD codes.

cs.IT↗

arXiv2026

Linear Programming Bounds for Locally Recovery Codes II

We give a polynomial-size linear programming bound for $q$-ary all-symbol locally recoverable codes with locality parameters $(r,δ)$, without assuming linearity. The key idea is to keep, for every ordered pair of codewords and every selected recovery view, the joint Hamming weight on the helper set, the recovered coordinate, and the rest of the code -- rather than collapsing this triple into a single distance, as earlier formulations do. Averaging this three-block distribution over recovery views of the same length yields exact identities linking it to the global distance distribution, together with nonnegative product-Krawtchouk constraints that encode locality and spectral positivity simultaneously. The resulting LP has polynomially many variables, its optimum dominates the ordinary Delsarte bound, and an earlier outside-distance formulation, the convex-hull bound of Li--Wei--Xiong, and the dual-based bound of Gruica--Jany--Ravagnani all arise from it as coarser marginals. Exact rational certificates over $q=2,3,4$ show the bound is strictly stronger than the best of these prior LPs in thirteen of fifteen tested cases, pinning down seven exact maximum code sizes and twelve exact maximum linear dimensions.

cs.IT↗

arXiv2026

Ramanujan Cayley Graphs with Normal Connection Sets in Ratio-One Frobenius Groups

Let $G=N\rtimes H$ be a finite Frobenius group with $|N|=q$ and $|H|=q-1$. We classify all Ramanujan Cayley graphs of $G$ whose connection sets are normal, in the sense of being unions of conjugacy classes. The group-theoretic input is a simple blow-up phenomenon: every such Cayley graph is either $Y[\overline{K_q}]$ or $Y[K_q]$ for a connected regular Cayley graph $Y$ on the complement $H$. We first prove a graph-theoretic result classifying all Ramanujan graphs of these two forms when $Y$ is an arbitrary connected regular graph on $q-1$ vertices. The proof combines the classical characterization of regular graphs with least eigenvalue greater than $-2$ with a second-moment identity in the bipartite case. Translating the resulting five graph types back to $G$ yields a complete classification for all ratio-one Frobenius groups, and in particular for $\operatorname{AGL}(1,q)$ over every finite field.

math.CO↗

arXiv2026

Hecke Algebra-valued Poincaré Series and Geometric Factorization of Affine Weyl Groups

This paper studies affine Weyl groups through the geometry of geodesic tubes and their hyperbolic stabilizers. We introduce a geometric framework relating affine Weyl group factorizations to Hecke-algebra-valued Poincaré series and zeta functions on quotient complexes. Using this framework, we prove a conjecture describing the connection between geodesic tubes and factorizations of Poincaré series for affine Weyl groups. We also establish partial results toward a Hecke-algebra-valued zeta identity for simply connected groups, including the cases of types $\widetilde{A}_{n-1}$ and $\widetilde{C}_n$. These results highlight the interaction between affine Weyl group geometry, Hecke algebras, and zeta functions arising from Bruhat--Tits buildings.

math.GR↗

arXiv2026

Information and Locality in Cayley Graphs

A de Bruijn sequence is the cyclic prototype of a Cayley-graph observation problem: when does the ordered label word on a translated window $gY$ determine the vertex $g$? We distinguish three parameters. The unrestricted number $\operatorname{sep}_q(G)$ minimizes an arbitrary separating pattern; the connected number $\operatorname{csep}_q(G,S)$ requires a connected Cayley window containing $Y_S=\{1\}\cup S$; and the one-step number $χ_1(G,S)$ fixes $Y_S$ and minimizes the alphabet. Thus $\operatorname{sep}_q$ is a group-level baseline, $\operatorname{csep}_q$ measures the cost of locality, and $χ_1$ tests the smallest prescribed local window. The organizing theme is the tension between information and locality. Carbon tori test the gap between $\operatorname{sep}_q$ and $\operatorname{csep}_q$: for generalized dihedral groups $\mathbb{F}_{\ell^d}^{\times}\rtimes C_2$ we prove, for odd prime powers $\ell$, the sharp baseline $\operatorname{sep}_\ell=d+1$ and construct connected zig-zag windows, while the order-$14$ Heawood torus satisfies $\operatorname{sep}_4=2$ and $\operatorname{csep}_4=4$. The spherical $A_5$ example and a finite simple-group comparison test the fixed one-step window: explicit symmetric cubic generating tuples give $χ_1(A_5,S)=3$ and $χ_1(\operatorname{PSL}_2(\mathbb{F}_7),S)=4$, both at the counting bound, with structured matrix-coefficient certificates. Cyclic-coset packings, finite-field coordinates, and restricted matrix coefficients are used only as the construction tools these two examples require.

math.CO↗

arXiv2026

Zeta functions of $\mathrm{PGL}_n$ over non-Archimedean local fields

Let $\mathscr{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_n(F)$, where $F$ is a non-Archimedean local field. We introduce geometric $k$-geodesics in $\mathscr{B}$ by means of CAT(0) convexity and combinatorial $k$-geodesics by a local successor relation on pointed $k$-facets. We prove that the two notions coincide. This allows us to use the local combinatorial definition on quotients $Γ\backslash\mathscr{B}$, without referring to the universal covering. When $Γ$ is discrete, torsion-free, cocompact, and type-preserving, the primitive closed $k$-geodesics define zeta functions $Z_k$ and their $ε$-twisted variants $Z_k^ε$. Our main result identifies an alternating product of these zeta functions with the unramified $L$-function of $L^2(Γ\backslash \mathrm{PGL}_n(F))$: $(1-u^n)^{χ(Γ\backslash\mathscr{B})}L(Γ,q^{(n-1)/2}u)=\prod_{k=1}^{n-1} Z_k^ε(Γ\backslash\mathscr{B},u)^{(-1)^{k+1}}$. This gives a uniform Ihara-type identity for all $\mathrm{PGL}_n$. We also extend the construction and the identity to $\mathrm{PGL}_n(D)$, where $D$ is a central division algebra over $F$; in that setting the residue parameter is $Q=|\mathcal{O}_D/\mathfrak{p}_D|$.

math.NT↗

arXiv2026

Universal Cycles for Affine Planes and 3-Subspaces over Finite Fields

We construct universal cycles for affine planes in $\mathbb F_q^n$ for all prime powers $q$ and all $n\ge4$, using sliding windows of length three. The construction is local-to-global: explicit local cycles are built on frame configurations, the linear $2$-subspaces are organized by a layered frame decomposition, and the resulting cycles are assembled by gluing along shared directions. The universal cycle obtained has direction set containing all $1$-subspaces. We also extend the construction to universal cycles for $3$-subspaces of $\mathbb F_q^n$.

math.CO↗

arXiv2026

From Characters to Matrices: An Elementary Construction of Irreducible Representations of Finite Groups

Let $G$ be a finite group and let $χ$ be an ordinary irreducible character. We give an elementary algorithm which constructs explicit matrices affording $χ$. The regular representation provides a canonical ambient representation, and the usual central idempotent projects onto the $χ$-isotypic component. The main step is then to maximize the squared norm of a diagonal matrix coefficient on the unit sphere of this component. The maximum is $1/χ(1)$, and it is attained precisely by vectors whose cyclic span is an irreducible subrepresentation affording $χ$. Thus the construction reduces the passage from characters to matrices to a concrete optimization problem. The same extraction method applies inside any smaller ambient representation containing $χ$, such as an induced representation from a subgroup. We complement the theory with a discussion on dimension reduction, robust numerical implementation, and an explicit $S_4$ example.

math.RT↗

arXiv2026

Universal Cycles on Affine Lines

A universal cycle is a cyclic sequence in which each object of a combinatorial family appears exactly once as a contiguous window. While such cycles are well understood for many discrete structures and linear subspaces, the case of affine lines presents additional difficulties arising from parallelism. We prove that universal cycles exist for affine lines in $\mathrm{AG}(n,q)$ for all $n \ge 2$ and all prime powers $q$. Our construction embeds the problem into $\mathrm{PG}(n,q)$, using points at infinity to encode directions, and proceeds via a decomposition into pairwise and triple configurations combined with a recursive lifting and gluing argument. We further interpret the construction in the Grassmannian $G_q(2,n+1)$, where affine lines correspond to the outer shell of $2$-subspaces, thereby extending known constructions for Grassmannians. A Python implementation is provided as supplementary material.

math.CO↗

arXiv2025

Spherical 2-Designs from Finite Group Orbits

We classify all spherical 2-designs that arise as orbits of finite group actions on real inner product spaces. Although it is well known that such designs can occur in representations without trivial components, we give a complete characterization of the orbits that satisfy the second-moment condition. In particular, we show that these orbits correspond to projections of compact group orbits within the regular representation, and we provide an explicit classification via isotypic decomposition and moment conditions. This approach unifies geometric and representation-theoretic viewpoints on highly symmetric point configurations.

math.CO↗

arXiv2019

Building lattices and zeta functions

We introduce the notion of a building lattice generalizing tree lattices. We give a Lefschetz formula and apply it to geometric zeta functions. We further generalize Bass's approach to Ihara zeta functions to the higher dimensional case of a building.

math.GR↗

arXiv2017

Zeta Functions of F1-buildings

The analogue of the Bruhat-Tits building of a p-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the p-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.

math.NT↗

arXiv2017

Geometric zeta functions for higher rank p-adic groups

The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions

math.NT↗

arXiv2017

Tree-lattice zeta functions and class numbers

The theory of Ihara zeta functions is extended to non-compact arithmetic quotients of Bruhat-Tits trees. This new zeta function turns out to be a rational function, despite the infinite-dimensional setting. In general it has zeros and poles, in contrast to the compact case. The determinant formulas of Bass and Ihara hold true if one defines the determinant as limit of all finite principal minors. From this analysis, a prime geodesic theorem is derived, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.

math.NT↗

arXiv2017

Zeta and L-functions of finite quotients of apartments and buildings

In this paper, we study relations between Langlands L-functions and zeta functions of geodesic walks and galleries for finite quotients of the apartments of G=PGL3 and PGSp4 over a nonarchimedean local field with q elements in its residue field. They give rise to an identity (Theorem 5.3) which can be regarded as a generalization of Ihara's theorem for finite quotients of the Bruhat-Tits trees. This identity is shown to agree with the q=1 version of the analogous identities for finite quotients of the building of G established in (KL1, KLW, FLW), verifying the philosophy of the field with one element by Tits. A new identity for finite quotients of the building of PGSp4 involving the standard $L$-function (Theorem 6.3), complementing the one in (FLW) which involves the spin L-function, is also obtained.

math.NT↗

arXiv2016

Twisted Poincare Series and Zeta functions on finite quotients of buildings

In the case where $G=$SL$_{2}(F)$ for a non-archimedean local field $F$ and $Γ$ is a discrete torsion-free cocompact subgroup of $G$, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat-Tits tree of $G$ by the action of $Γ$, and an alternating product of determinants of twisted Poincaré series for parabolic subgroups of the affine Weyl group of $G$. We show how this can be generalised to other split simple algebraic groups of rank two over $F$, and formulate a conjecture about how this might be generalised to groups of higher rank.

math.GR↗

arXiv2013

On spherical Monte Carlo simulations for multivariate normal probabilities

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. This paper proposes a spherical Monte Carlo method with both theoretical analysis and numerical simulation. First, the multivariate normal probability is rewritten via an inner radial integral and an outer spherical integral by the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo experiments on some probabilities calculation confirm these claims.

stat.CO↗