Papers

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the length of the unique cycle of $G$. For the adjacency eigenvalues of $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of the positive and negative eigenvalues, respectively. Akbari, Kumar, Mohar, Pragada, and Zhang conjectured that, when $k$ is odd, the value of $k$ modulo $4$ determines which of $s^+(G)$ and $s^-(G)$ is greater than $n$. More precisely, if $k\equiv 3\pmod 4$, then $s^+(G)>n>s^-(G)$; if $k\equiv 1\pmod 4$, then $s^+(G)<n<s^-(G)$. We confirm this conjecture.

For a commutative ring $R$ with identity, the weakly zero-divisor graph $W\Gamma(R)$ has vertex set $Z(R)^{\ast}$, with distinct vertices $x$ and $y$ adjacent whenever there exist nonzero $r\in{\rm Ann}(x)$ and $s\in{\rm Ann}(y)$ with $rs=0$. The Laplacian spectrum of $W\Gamma(Z_n)$ has been determined by Shariq, Mathil, and Kumar, who also established that $W\Gamma(Z_n)$ is Laplacian integral. Building on the structural description of $W\Gamma(R)$ due to Nikmehr, Azadi, and Nikandish, we extend the Laplacian spectrum and integrality results from $Z_n$ to every finite commutative ring $R$: we restate $W\Gamma(R)$ in unified form as a complete multipartite graph whose parts are made explicit by the local-ring decomposition of $R$, compute the full Laplacian spectrum in closed form, prove Laplacian integrality of $W\Gamma(R)$, and give a sharp bound on the number of distinct Laplacian eigenvalues. As consequences we obtain explicit formulas for the algebraic connectivity and number of spanning trees of $W\Gamma(R)$, and recover the Laplacian spectrum of $W\Gamma(Z_n)$ in compact form.

The second Eulerian numbers are defined via the descent enumerator of Stirling permutations, a class of permutations introduced by Gessel and Stanley. We give a simple and conceptual proof of two identities relating the Bernoulli numbers and the second Eulerian numbers. We rely on a recent Ehrhart-theoretic idea of Ferroni.

In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer than $d$ common colors. The $t$-tone chromatic index of $G$, denoted by $\tau'_t(G)$, is the minimum integer $k$ for which $G$ admits a $t$-tone edge $k$-coloring. We focus on the case $t=2$ and establish several upper bounds on $\tau'_2$. In particular, for every graph $G$ with maximum degree $\Delta(G)\ge2$, we prove that $\tau'_2(G)\le 6\Delta(G)-4$, improving the corresponding bound derived from the vertex analogue. We also show that every tree $T$ with $\Delta(T)\ge3$ satisfies $\tau'_2(T)=2\Delta(T)$. Furthermore, every planar graph $G$ satisfies $\tau'_2(G)\le \max\{41,3\Delta(G)+5\}$, while every outerplanar graph $G$ satisfies $\tau'_2(G)\le \max\{14,3\Delta(G)\}$. For subcubic graphs $G$, the vertex analogue yields $\tau'_2(G)\le12$. We improve this bound to $11$ for claw-free subcubic graphs and to $10$ for $2$-degenerate subcubic graphs. Finally, we propose two conjectures concerning optimal bounds for cubic and $K_4$-free cubic graphs, and establish them for series-parallel subcubic multigraphs and subcubic outerplanar graphs, respectively.

Pipedreams and bumpless pipedreams are two combinatorial models that compute double Grothendieck polynomials. While studying matrix Schubert varieties, Pechen Pipedreams and marked bumpless pipedreams are two combinatorial models that compute double Grothendieck polynomials. While studying matrix Schubert varieties, Pechenik, Speyer, and Weigandt defined the Rajchgot code, denoted by $\mathsf{rajcode}(\cdot)$, which captures the leading monomial of the top-degree component of a Grothendieck polynomial. Combinatorially, their result implies that there exists a unique pipedream (or marked bumpless pipedream) with row weight $\mathsf{rajcode}(w)$ and column weight $\mathsf{rajcode}(w^{-1})$. A construction of such a pipedream was subsequently given by Chou and Yu. In this paper, we resolve the marked bumpless pipedream version of this problem by providing an explicit algorithm.

Among the many proofs of Turán's classical theorem, one particularly surprising proof due to Li and Li uses ideals in polynomial rings to record missing edges. Motivated by their proof, Mubayi proposed a hypergraph analogue, conjecturing that an ideal generated by multipartite $3$-graphs coincides with a differentiated diagonal-vanishing ideal. If true, this conjecture would imply the extremal-number part of Mubayi's classical hypergraph extension of Turán's theorem. We disprove this conjecture throughout the nontrivial parameter range. We then give an alternative algebraic proof of Mubayi's extremal formula for the family $\mathcal{K}_{\ell}^{(r)}$ of clique expansions, using monomial cover ideals and a Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients.

We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.

This paper studies generalized Pólya conversion problems for the $q$-permanent. We establish a sharp threshold governing the transition from low-dimensional algebraic flexibility to higher-dimensional combinatorial rigidity. For $n \ge 3$ and $q \ne \pm 1$, we prove that the $q$-permanent is not linearly convertible to the determinant or permanent. Conversely, we completely classify the space of Schur multiplier preservers for $n=2$. Focusing on Schur multipliers, we characterize the preserver exponents as a $(2n-2)$-dimensional space of additive matrices. We show that for lower Hessenberg matrices, the general geometric obstruction disappears, yielding an explicit determinantal reduction and an $\mathcal{O}(n^3)$ evaluation algorithm. Furthermore, we classify permutational converter exponents, proving that for $n \ge 4$, admissible symmetries are strictly constrained to the dihedral group. Finally, we resolve a mixed conversion problem, showing the solution space is nonempty only for $n \le 4$, which provides a direct algebraic characterization of the $q$-permanent's zero locus in low dimensions.

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a “universal generating set” for the module of splines over any graph with fixed combinatorial genus. This theorem holds over any Noetherian commutative ring with a chosen finite list of ideals for edge-labels. We then give several applications of this theorem, including showing that a particular generating function associated to splines on trees is algebraic when the base ring satisfies certain finiteness conditions. We illustrate our technical theorems explicitly by giving a classification of splines on graphs with combinatorial genus one and two.

Framing triangulations of unit flow polytopes have received a great deal of recent study with rich connections to various generalizations of Catalan and Cambrian combinatorics as well as volume and h*-polynomial formulas. This story has largely been restricted to unit flow polytopes, with only two recent works giving descriptions of framing triangulations on classes of non-unit flow polytopes. In this article we introduce the first theory of (unimodular) framing triangulations for arbitrary integer flow polytopes. We will observe some pathologies in general examples which are impossible in the unit case, and propose in response a class of "well-ordered" framing triangulations which we expect to inherit key properties from the unit case while still containing all settings of framing triangulations existing in the literature.

In the study of flow polytopes, a directed acyclic graph (DAG) with a choice of framing gives a regular unimodular triangulation on its space of unit nonnegative flows. In representation theory, a gentle algebra has recently been equipped with a space of unit flows admitting triangulation and subdivision results capturing its tau-tilting theory. These theories from different areas of mathematics overlap: flows on gently framed DAGs are the same as flows on paired representation-finite gentle algebras. In this article we develop the common generalization of these two theories by defining (framed) turbulence charts, which may be thought of as analogs of (framed) DAGs without the conditions of (D)irectedness and (A)cyclicity. The space of unit flows on a turbulence chart is its turbulence polyhedron. We give presentation, subdivision, and triangulation results on turbulence polyhedra which restrict to known results in the settings of framed DAGs and gentle algebras.

Boolean networks are widely used to model gene regulatory systems. Their structural and dynamical properties are commonly interpreted by comparison with ensembles of random Boolean networks generated by sampling Boolean functions for individual nodes. Canalizing and nested canalizing functions, in which one or more regulatory inputs dominate the output, capture an important feature of gene regulation. These functions are typically generated by sampling their defining parameters uniformly at random. Because multiple parameterizations can represent the same Boolean function, however, this procedure induces a biased distribution over functions and consequently over null models. We develop efficient algorithms for uniformly sampling Boolean functions with prescribed canalizing depth, thereby correcting this systematic bias. Using these unbiased null models, we show that the sampling measure substantially alters function- and network-level properties. Whereas parameter-uniform sampling yields nested canalizing functions with expected average sensitivity one, these sensitivities increase with degree under function-uniform sampling and approach 1.183. These differences alter expectations for robustness, attractor structure, and stability. Reanalysis of 122 published Boolean gene regulatory network models reveals substantially stronger enrichment of low-sensitivity canalizing architectures than previously recognized. Widely used parameter-based null models therefore systematically underestimate baseline sensitivity and overestimate the stabilizing role of canalization.

For an integer $t \geq 1$, a homomorphism of a digraph G to a digraph $H$ is $t$-frugal if no more than $t$ in-neighbours of any vertex of $G$ have the same image. There is a dichotomy theorem based on structural properties when $t=1$ and $H$ has a loop at each vertex, but there is unlikely to be such a theorem for general digraphs $H$. For $t \geq 2$ we describe a structural dichotomy theorem for $t$-frugal homomorphisms of general digraphs.

Young tableaux are fundamental objects in algebraic combinatorics and representation theory, with operations such as promotion and jeu de taquin playing a central role in their structure and applications. While these operations are well understood for finite tableaux, their behavior on infinite tableaux has so far been studied mainly within probabilistic frameworks. In this paper, we investigate jeu de taquin on infinite standard Young tableaux from a purely combinatorial and dynamical point of view. We analyze the action of jeu de taquin on infinite shapes, describe the structure of inverse images, and classify tableaux exhibiting periodic, pre-periodic, and recurrent behavior. We also introduce a natural metric on the space of infinite tableaux and show that jeu de taquin defines a chaotic dynamical system in the sense of Devaney. These results extend classical tableau theory to infinite settings and identify connections between combinatorial dynamics and infinite representation-theoretic structures.

In this paper, we complete the enumeration of the number of parking functions of length $n$ avoiding, in the sense defined by Qiu and Remmel, a permutation of length 3, answering several questions of Adeniran and Pudwell. Additionally, we provide explicit growth rates for the number of parking functions of length $n$ avoiding a monotonic pattern of any length. As a consequence of our techniques we additionally provide a combinatorial enumeration for the number of equivalence classes of words with a fixed content under the Sylvester congruence.

We study the ideal of the algebraic relations among 3-point functions from a combinatorial and topological perspective. We place this problem in the broader setting of incidence toric ideals associated with incidence matrices of t-subsets contained in k-subsets of n elements. Generators of these ideals admit combinatorial interpretations as null t-designs and topological interpretations as balanced orientable normal d-pseudomanifolds without boundary. Generators arising from octahedra play a fundamental role in the structure of these ideals.

Chomp was introduced by Gale in 1974 \cite{Gale1974}. In the same paper, Gale reported that the $3\times n$ games had been completely analyzed for $n\le 100$, with a unique winning first move in every case, and asked whether winning first moves are unique in general. Although the general uniqueness statement is false \cite[Section~7.1]{BrouwerEtAl2005}, we prove that the three-row uniqueness phenomenon suggested by Gale's computations holds for all $n$: every $3\times n$ Chomp rectangle has exactly one winning opening move. This settles the three-row case of Gale's 52-year-old first-move uniqueness question. The proof is carried out in the two-variable recurrence introduced by Brouwer, Horváth, Molnár-Sáska, and Szabó \cite{BrouwerEtAl2005} for the function $f(q,r)$ whose values encode the $P$-positions. The main local ingredient is a rightmost-hole principle: if a value $p$ is absent from the set $C(q,r)$ but belongs to all corresponding sets $C(t,r)$ for $q<t<p$, then all intermediate values $q+1,\ldots,p-1$ are forced to belong to $C(q,r)$. This separates the diagonal values from the starts of constant rows, and yields a partition of the positive integers into the two possible types of winning opening moves. It also identifies the row of the unique opening move: no first-row opening move is winning; the second-row and third-row cases are precisely the two complementary Chomp sequences A029900 and A029901.

Motivated by majority vertex-colorings of graphs and digraphs and majority edge-colorings of graphs, we introduce two concepts of strong majority colorings. A strong majority vertex-coloring of a graph $G=(V,E)$ is a mapping $c:V\rightarrow C$ such that for every vertex $v\in V$ and every color $\alpha\in C$, at most half of the neighbors of $v$ have color $\alpha$. The strong majority number of $G$, denoted Maj$(G)$, is the least number of colors in such a coloring. We show that Maj$(G)$ can be arbitrarily large and prove a tight upper bound Maj$(G)\le 2\Delta(G)+1$ for every graph $G$ without pendant vertices. A strong majority edge-coloring of a graph $G$ is a mapping $c:E\rightarrow C$ such that for every edge $e\in E$ and every color $\alpha\in C$, at most half of the edges adjacent to $e$ have color $\alpha$. The strong majority index of $G$, denoted Maj'$(G)$, is the least number of colors in such a coloring. It is shown that there is an upper constant bound for Maj'$(G)$ of all admissible graphs $G$. We conjecture that this constant is as small as 4 and confirm this conjecture for numerous graph classes.

We determine which connected sets of $N$ vertices in the square lattice support the most spanning trees, using all lattice edges between the chosen vertices. Every maximizer minimizes the number of edges joining the set to the rest of the lattice. Its shape is a rectangle with at most two straight strips of vertices removed at opposite corners. For $N\ge18$, the maximizer is unique up to translation, rotation, and reflection: an explicitly determined rectangle with one possibly empty corner strip removed along its longer side. The proof combines comparisons of Laplacian determinants with geometric comparisons of corner deletions. For $N=n^2$, we prove a sharp stability inequality that bounds the symmetric difference from a translated $n\times n$ square in terms of the loss in logarithmic tree count. Thus, nearly optimal tree counts force the vertex set to be close to a square. We also derive an exact product formula and a recurrence for the number of spanning trees in rectangles with a single corner strip removed.

For a graph $G$, the spectral radius of $G$ is the largest eigenvalue of its adjacency matrix. A connected factor of $G$ is a connected spanning subgraph of $G$. For example, a spanning tree of $G$ is a 1-connected factor of $G$. Let $G$ be a graph of order $n$ with minimum degree $\delta\geq6$, where $n\geq3\delta$. In this paper, we give a sharp spectral radius condition for $G$ to contain $k$ edge-disjoint 2-connected factors and $\left\lfloor\frac{\delta-4k}{2}\right\rfloor$ edge-disjoint spanning trees, where $1\leq k\leq\left\lfloor\frac{\delta}{4}\right\rfloor$ is an integer.

In this paper we completely classify semifields of order $2^8=256$ containing a nucleus of order $2^4=16$. We introduce new invariants for semifields, and apply new computational techniques for calculating old invariants. Together these make the computational classification significantly quicker.

Graph coloring with preferences offers a powerful framework for constraint satisfaction problems in which fulfilling every request is impossible but satisfying a guaranteed positive fraction is highly desirable. A request on a graph $G$ equipped with a list assignment $L$ assigns to each vertex of some subset $dom(r)\subseteq V(G)$ a preferred color from its list. Following Dvořák, Norin, and Postle (2019), $G$ is \emph{$\varepsilon$-flexibly $k$-choosable} if, for every $k$-list assignment $L$ and every request $r$, there is an $L$-coloring of $G$ that agrees with $r$ on at least $\varepsilon|dom(r)|$ vertices. The corresponding notion for DP-coloring (correspondence coloring) was formalized by Bradshaw, Choi, and Kostochka (2025). Choi, Clemen, Ferrara, Horn, Ma, and Masařík (2022) proved that every planar graph without $4$-cycles and with $3$-cycle distance at least $2$ is $\varepsilon$-flexibly $4$-choosable. We improve the result in two respects: weakening the hypothesis from $3$-cycle distance $\geq 2$ to vertex-disjoint triangles, and strengthening the conclusion from list flexibility to weighted DP-flexibility: \emph{Every simple planar graph without $4$-cycles and without intersecting triangles is weighted $\varepsilon$-flexibly DP-$4$-colorable.} The list size $4$ is sharp: Montassier, Raspaud, and Wang constructed a planar graph without $4$-cycles, $5$-cycles, and intersecting triangles that is not $3$-choosable.

We study intersection size distributions arising in finite geometries and vector spaces. For a projective plane $\Pi_q$ of order $q$ with line set $\mathcal L$, we show that \[ \min_{S\subseteq\Pi_q}\max_k |\{\ell\in\mathcal L:|S\cap\ell|=k\}|=\Theta(q^{3/2}). \] The arguments behind these bounds admit a more general formulation. We prove a similar lower bound for balanced incomplete block designs, together with probabilistic upper bounds for uniform block systems: one for fixed block size under a weak dependency condition and another for growing block size under bounded pairwise intersections. This yields corresponding results for subspaces of vector spaces over finite fields, equivalently, for affine subspaces of an affine space. We also study explicit constructions in affine planes of prime order and relate their intersection size distributions to character sum estimates; an elliptic curve construction gives a bound within a polylogarithmic factor of the optimal order. Finally, we relate balanced intersection size distributions to legitimate colorings and prove that every n-uniform linear hypergraph with $n$ edges admits a legitimate 2-coloring.

The problem of computing the cardinality of the intersection of multiple balls in the Hamming space has attracted a lot of attention recently due to their applications in the list reconstruction problem and information retrieval in Associative Memories. In previous work, most of the results are for the cases where the radii of each ball, $r$ and the distance between the centers of these balls, $k$ are fixed when the length $n$ of each codeword tend to infinity. In this work, we focus on the case where $r = \alpha n$ and $k=\beta n$ for some constants $\alpha$ and $\beta$ and compute the maximum asymptotic rate of the cardinality of the intersection of three balls. We provide the maximum asymptotic rate as a function of two parameters $\alpha$ and $\beta$. We also provide numerical results and compare these results with the intersection of two balls.

Why does the low dimensionality of representations, typically $d\approx 1000$, not prevent modern embedding-based retrieval models from scaling to billions, or even trillions, of data points? To answer this question, we study maximal-margin embeddings in the following retrieval model, classically studied in communication complexity [PS86] and more recently in embedding-based retrieval [WBNL26]. Let $A\in \{0,1\}^{N\times n}$ be a matrix indicating whether each of $N$ queries is relevant to each of $n$ documents. We are interested in the largest margin $m>0,$ denoted by $\mathsf{m}^{\mathsf{rd}}(d, A),$ for which there exist unit norm embeddings of the queries and documents $\{U_j\}_{j = 1}^N, \{V_i\}_{i = 1}^n$ with the following property. $\langle U_j, V_i\rangle \ge m$ whenever $A_{ji} = 1$ and $\langle U_j, V_i\rangle \le -m$ otherwise. A large margin is a key proxy for representation quality: it controls both robustness to perturbations and compositional generalization across queries. Our main theorem establishes that the best possible margin without a restriction on the dimension, $\mathsf{m}^{\mathsf{rd}}(+\infty, A),$ can be nearly achieved in dimension $d = O(\mathsf{m}^{\mathsf{rd}}(+\infty, A)^{-2}\log n)$ which improves a theorem of [BDES02]. Together with a matching lower bound in Theorem 1.5, we conclude that when $A\in \{0,1\}^{\binom{n}{k}\times n}$ is the matrix containing all possible $k$-sparse rows once, dimension $d = O(k\log (n/k))$ is necessary and sufficient for the maximal possible margin $\mathsf{m}^{\mathsf{rd}}(+\infty, A) = \Theta(k^{-1/2})$ in this setting. This fully resolves the setup of [WBNL26]. We also give several constructions for large margins when $d = o(k\log (n/k)).$ Finally, we empirically test the InfoNCE and sigmoid losses for producing large margin embeddings and demonstrate a clear advantage of the sigmoid loss.

Our focus is upon irreducible nonnegative $n$-by-$n$ matrix realizations of nonnegatively realizable spectra or, equivalently, characteristic polynomials. After giving some general background, we make some useful new observations and show the existence of irreducible nonnegative realizations in some general cases. Then, we focus on $n<5$, where the NIEP is solved. Finally, we focus on the trace 0 case and, using graph theoretic methods, characterize nonnegative irreducible realizability among realizable polynomials. The closely related problem of positive realizations, for trace positive spectra, is also discussed.

In this paper, we introduce the notion of Pfaffian orientations on (punctured) polygonally cellulated orientable surfaces, and provide an expression for the number of such orientations. This generalizes the notion of Pfaffian orientations on planar graph, where a planar graph is seen as a punctured $2$-sphere, embedded in $\mathbb{R}^3$. So, as a direct corollary of our main theorem, we derive the number of Pfaffian orientations on a planar graph.

In Ehrhart theory, the well-known sign pattern problem asks: given a positive integer $d\geq 3$ and integers $1 \leq i_1 < \cdots < i_k \leq d-2$, does there exist a $d$-dimensional integral polytope $\mathcal{P}$ such that in its Ehrhart polynomial $i(\mathcal{P}, t)$ the coefficients of $t^{i_1}, \ldots, t^{i_k}$ are negative, while all remaining coefficients are positive? This problem was proposed by Hibi, Higashitani, Tsuchiya, and Yoshida. In this paper, we first construct a class of simplices $\mathcal{S}_d(m)$ whose Ehrhart polynomial has leading coefficient $m$ and all other coefficients fixed positive constants. Then, using the Cartesian product of $\mathcal{S}_d(m)$ and the Reeve tetrahedron, we obtain the first complete solution to the sign pattern problem. Finally, while attacking the sign pattern problem, we discovered a fast algorithm for computing the $h^*$-polynomial of a class of simplices $\Delta(0,q)$. This algorithm is crucial for constructing the simplices $\mathcal{S}_d(m)$.

For a fixed integer $m\ge 1$, let $\mathcal{A}_n^{(m)}$ be the set of permutations $\pi\in S_n$ that avoid the pattern $132$ and satisfy the adjacency bound $|\pi_{i+1}-\pi_i|\le m$ for all $i$. Here, a pattern $132$ means three indices $i<j<k$ such that $\pi_i<\pi_k<\pi_j$. A recent study initiated the enumeration of these constrained 132-avoiding permutations, treating the case $m=2$ by deriving a rational ordinary generating function and asking for finite-state decompositions, rational generating functions, and explicit rational formulas for larger fixed $m$. We introduce a two-sided endpoint-state decomposition that works uniformly for every fixed $m$. The state variables impose threshold bounds on the endpoint deficiencies $n-\pi_1$ and $n-\pi_n$, with thresholds in $\{0,1,\ldots,m-1,\infty\}$. This gives at most $(m+1)^2$ states and proves that, for every fixed $m$, the ordinary generating function $A^{(m)}(x)$ is rational and can be computed effectively by exact linear algebra. We also identify cyclic strongly connected components of the dependency graph in the finite-state system to give an explicit upper bound for the order of an eventual constant-coefficient recurrence satisfied by the sequence $a_n^{(m)}=|\mathcal{A}_n^{(m)}|$. We then recover the known case $m=2$ from this state system and work out the case $m=3$ explicitly. On the asymptotic side, we prove that the exponential growth constant exists for every $m$; for $m\ge2$ it is obtained from the spectral radii of the two cyclic components with more than one vertex in the state system. We determine the simple-pole asymptotics for $m=2$ and $m=3$, and we prove that the growth constants are nondecreasing in $m$, strictly smaller than the Catalan growth constant $4$ for every finite $m$, and converge to $4$ as $m\to\infty$.

This paper provides a comprehensive analysis of almost maximum distance separable (AMDS) constacyclic codes of length $4p^{\varsigma}$ over the finite field $\mathbb{F}_{p^m}$, where $p$ is an odd prime number. Furthermore, it introduces the construction of quantum AMDS (qAMDS) codes, drawing on the principles of the Calderbank-Shor-Steane (CSS) framework, which enhances their applicability in quantum error correction. This work aims to deepen the understanding of these codes and their potential uses in both classical and quantum computing environments.