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arXiv2026

Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

math.CO↗

arXiv2026

Further Comments on Yablo's Construction

We continue our analysis of Yablo's coding of the liar paradox by infinite acyclic graphs. The present notes are based on and continue the author's previous results on the problem. In particular, our approach is often more systematic than before.

math.CO↗

arXiv2026

It's Hard to PArcK

We show that Partizan Arc Kayles (PArcK), a generalization of Domineering to graphs, is PSPACE-complete via a reduction from Positive CNF and with recently-discovered techniques for creating PArcK positions with high temperature. The reduction uses only red and blue edges.

cs.CC↗

arXiv2026

Unfolding Overlaps of the Exceptional Regular Polytopes

We find explicit ridge unfoldings of the three exceptional 4D polytopes (24-cell, 120-cell, 600-cell) that result in overlaps of their facets. These failures bring an end to the full classification of regular polytopes with the all-net property.

cs.CG↗

arXiv2026

Double Toeplitz codes and their average weight enumerators

Recently, double Toeplitz codes have been introduced as a generalization of double circulant codes. In this paper, we study the average weight enumerators of double Toeplitz codes. As an application, we consider the existence of double Toeplitz codes over $\mathbb{F}_q$ with some specified minimum weights for $q \in \{2,3,4\}$. We also give a classification of double Toeplitz codes over $\mathbb{F}_q$ with the largest minimum weights for moderate lengths and $q \in \{2,3,4\}$.

math.CO↗

arXiv2026

Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices

In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \times m$ minors equals $Δ$.

math.CO↗

arXiv2026

Edge codes constructed from unicyclic graphs

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

math.CO↗

arXiv2026

Distinguishing classes of intersection graphs of homothets or similarities of two convex disks

For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior and with at most one tangent at every boundary point, we classify the classes $G^{\text{hom}}(A)$ and $G^{\text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities of $A$, respectively. Namely, we prove that $G^{\text{hom}}(A)=G^{\text{hom}}(B)$ if and only if $A$ and $B$ are affine equivalent, and $G^{\text{sim}}(A)=G^{\text{sim}}(B)$ if and only if $A$ and $B$ are similar.

cs.CG↗

arXiv2026

An Exposition of the $\widetilde{O}(\log^{1/4} n)$ Bound for the Komlós Problem

A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $Ω(\sqrt{\log n})$ should hold.

math.CO↗

arXiv2026

Covering 1024 syndromes with 50 columns

We exhibit a binary linear $[50,40]_2$ code of covering radius $2$, so $\ell_2(10,2)\le 50$, one column below the Kaikkonen--Rosendahl length $51$ that has stood since 2003 and that still seeds the $R=2$ family of Davydov--Marcugini--Pambianco (arXiv:2511.02542). The new matrix admits a $(2,0)$-partition into ten blocks, so Construction $\mathrm{QM}_2^2$ propagates it to exhaustively verified codes of lengths $815$ and $1631$ at $r=18$ and $r=20$, and to the family $n=51\cdot 2^{r/2-5}-1$ of asymptotic density $2601/2048$. The matrices, verifiers, and source are at https://github.com/wustep/maths, pin problems/covering/share/2026-08-24/ at commit 736a38f.

cs.IT↗

arXiv2026

Efficient Polynomial-Time Decoding of Simplicial Anticodes with Near-Optimal Performance

In this work, we propose an efficient decoding algorithm for codes arising from simplicial complexes, a family of binary linear codes for which no decoding method of this type was previously known. Although the algorithm does not always attain the maximum theoretical error-correcting capability, it provides an explicit bound that can be computed directly from the structure of the complex. Moreover, this bound is asymptotically optimal: the ratio between the guaranteed correcting capability and the theoretical maximum converges to $1$ as the code length increases, under natural assumptions on the dimension of the maximal faces. The correction capability is also presented in specific examples. Finally, we introduce specific families of simplicial complexes where the algorithm successfully reaches this theoretical bound.

cs.IT↗

arXiv2026

An Improved Bound for Smith's Longest Cycles Conjecture via a Forbidden Subdivision

Smith's conjecture asserts that in every $k$-connected graph with $k\geq 2$, any two longest cycles intersect in at least $k$ vertices. In this work, we establish an $Ω(k^{8/11})$ bound for this conjecture, improving upon the $Ω(k^{2/3})$ bound of Ma and Zhao. Our proof combines a Ramsey theoretic refinement of the traditional Turán-type approach with computer search.

math.CO↗

arXiv2026

Discrepancy of geometric incidences

We study the combinatorial (red-blue) discrepancy of finite point sets with respect to hyperplanes and, more generally, bounded-complexity affine algebraic sets. We prove that every $n$-point set in a real Euclidean space admits a red-blue coloring for which every affine algebraic set of dimension at most $D$ and degree at most $k$ has discrepancy at most $n^{\frac12-\frac{1}{2(D+1)}-\varepsilon}$ for some $\varepsilon=\varepsilon(D,k)>0$. This gives a polynomial improvement over the straightforward VC-dimension bound $\tilde O(n^{\frac12-\frac{1}{2(D+1)}})$. In the opposite direction, we construct $n$-point sets in $\mathbb R^d$ whose discrepancy with respect to hyperplanes is $\tildeΩ(n^{\frac12-\frac{1}{d+1}}),$ extending the point-line discrepancy lower bound of Chazelle and Lvov. We present further applications of our methods in communication complexity, concerning separation between randomized communication cost and deterministic communication cost with access to equality oracle.

math.CO↗

arXiv2026

Entropy lower bounds and sum-product phenomena

Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (Rényi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$.

math.CO↗

arXiv2026

On two proofs of $d^2$ mixing of weighted Dikin walks

We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, $\barν$-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an $\widetilde O(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\widetilde O(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger $χ^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an $\widetilde O(d^2)$ mixing bound in $χ^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.

cs.DS↗

arXiv2026

Fine Difference Structure and Prime-Power Depth of Bent Partitions

A $p$-ary bent partition of $\mathbb{F}_p^n$ is a partition into $K$ nonempty cells such that every balanced assignment of its cells to $\mathbb{F}_p$ produces a bent function. It was asked whether every possible depth $K$ is a power of $p$; for general $p$, previous affirmative results required regularity or cell-symmetry hypotheses. We prove the stronger unconditional statement that, for every nonzero $h$, exactly $p^n/K$ points remain in the same fine cell under translation by $h$. Thus the fine cells form a partitioned difference family and the fine label map is zero-difference balanced. Consequently $K\mid p^n$, so $K=p^t$; nonempty cells further give $1\le t<n$. In even dimension, the classical cell-size theorem yields $K\mid p^{n/2}$. Together with the known odd-dimensional ternary three-fibre parameter restriction, this gives the global bound $t\le\lfloor n/2\rfloor$. The proof is an exact finite average over balanced coarsenings. The main counting identity and selected consequences are formalized and kernel-checked in Lean 4.

cs.IT↗

arXiv2026

Towards a mathematical theory of superposition

We develop a mathematical theory of superposition in neural networks using tools from frame theory and compressed sensing. In our model, a sparse binary vector $x$ of active features is encoded through an overcomplete dictionary $W$, and feature recovery is performed by applying $\operatorname{ReLU}(W^\top W x+b)$ with an appropriate bias vector $b$. We prove several recovery theorems for this model. In the random-support setting, we establish high-probability support recovery for nearly tight, low-coherence dictionaries, with guarantees when the expected sparsity is up to order $d/\log n$. In the worst-case support setting, we give a sharp and computable criterion for which sparsity levels permit support recovery. We apply this criterion to Gaussian random matrices and equiangular tight frames. For real equiangular tight frames with $n>d+1$, we determine the exact recovery threshold in terms of the coherence. The proof of this result for real equiangular tight frames relies on a novel characterization---which should be of independent interest to frame theorists---of the distribution of signs in the Gram matrix.

stat.ML↗

arXiv2025

The differential properties of certain permutation polynomials over finite fields

Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their varied applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown by Anbar et al. (J. Comb. Des. 31(12):1-24, 2023). Additionally, in a recent manuscript by Pal et al. (Adv. Math. Communications, to appear), a very interesting connection between the $c$-differential uniformity and boomerang uniformity, when $c=-1$, was pointed out, showing that they are the same for an odd APN permutation, sparking yet more interest in the construction of functions with low $c$-differential uniformity. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic~$3$. To solve the involved equations over finite fields, we use various number theoretical techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest.

math.CO↗