It is demonstrated that local subgraph statistics alone are insufficient to surpass the GV bound in the Hamming case, suggesting that improvements must stem from large-scale structural properties of the space.
Abstract
This paper investigates the relationship between coding theory and extremal combinatorics by representing codes in general metric spaces as independent sets in proximity graphs. We provide a generalized framework for the Gilbert-Varshamov (GV) bound applicable to codes over any finite metric space and explore the conditions under which global combinatorial parameters can force the existence of codes exceeding this bound. Central to our analysis is the introduction of Ramsey-Sidorenko and independence-forcing graphs. We establish density thresholds for various graph families and utilize the Karush--Kuhn--Tucker conditions to analyze entropy optimization in the Hamming case. Furthermore, we derive upper bounds on code sizes using fractional packings in vertex-transitive and nonedge-transitive graphs. Our findings demonstrate that local subgraph statistics alone are insufficient to surpass the GV bound in the Hamming case, suggesting that improvements must stem from large-scale structural properties of the space.
We study upper bounds on the length of $\mathbb F_q$-linear QMDS codes in the folded Hamming distance relative to their other parameters, especially the field size $q$. Via a correspondence between such codes and families of subspaces, we relate the length problem to that of upper bounding $1$-subspace packings with respect to the other parameters, especially the field size. Our main result is a reduction from these families to partial spreads, which allows us to import sharp bounds from finite geometry, including results of Drake-Freeman, N\u{a}stase-Sissokho, and Honold-Kiermaier-Kurz. As a consequence, we recover the Griesmer-type upper bound on the length of QMDS codes by Ball et al. and obtain tighter upper bounds in several parameter regimes.
The Ramsey number $R(m,n)$ is the smallest order at which every red-blue edge coloring of a complete graph must contain a blue clique (a complete subgraph) of size $m$ or a red clique of size $n$. Determining these numbers exactly is extremely hard, and even certifying a lower bound requires exhibiting an explicit coloring that avoids both cliques. We develop an integer programming framework for certifying such lower bounds, restricting the search to circulant graphs, whose rotational symmetry lets us reformulate the problem in a projected distance space, reducing the number of binary variables from quadratic to linear in the graph order. We strengthen this projected model through coefficient reduction and solve it with a branch-and-cut algorithm whose separation routine exploits the common neighborhood structure of circulant graphs, combining heuristic and exact maximum-clique algorithms. In an extensive computational campaign on circulant graphs with up to 410 vertices, we improve the best lower bounds previously obtained by other methods by up to 11 points for 25 values of $R(3,n)$ with $24\le n\le49$ and $n\neq27$, each backed by an explicit graph certificate that can be independently verified with a stand-alone exact clique solver. To the best of our knowledge, our method also provides the first reproducible optimization-based procedure for certifying circulant Ramsey numbers $R_C(m,n)$, which we use to establish eight new values of $R_C(3,n)$ with $13\le n\le20$. Our framework, graph certificates, and stand-alone checker are provided as supplementary material to support independent verification and reuse.
Stefano Coniglio, Fabio Furini, I. Ljubić et al.· 0 citations
We study Schubert subspace codes, which are constant-dimension subspace codes with prescribed intersection conditions with a fixed subspace. Our goal is to construct codes of maximum possible size in the extremal distance cases where a natural counting upper bound applies. We give two families of constructions. The first one uses a direct-sum decomposition of the ambient space, together with partial spreads and colorings of powers of $q$-Johnson graphs. For this construction, we also prove necessary conditions, which show how chromatic and clique obstructions arise. The second family is obtained by field reduction from evasive and scattered subspaces over extension fields. This gives codes whose size can be computed exactly in the scattered case and recovers the only previously known construction as a special case.
Gianira N. Alfarano, Alessandro Neri, Beatrice Toesca· 0 citations
Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.
Vishal Gupta, A. Iosevich, J. Iosevich et al.· 0 citations
Torquato and Stillinger conjectured an exponential improvement of Minkowski's classical lower bound on the maximal density of sphere packings in high-dimensional Euclidean space $\mathbb{R}^d$ using a pair-correlation-function optimization framework. Conditional on their realizability conjecture, we show that a simple family of hyperuniform pair correlation functions yields polynomial improvements over Minkowski's lower bound of the form $\phi_{\mathrm{max}} \gtrsim d^\beta 2^{-d}$ for every fixed $\beta>1$ in sufficiently high dimensions. As the polynomial exponent is allowed to increase with dimension, this family continuously approaches the previously conjectured exponential improvement. We further derive the same exponential asymptotic rate independently from the Cohn--Elkies dual linear programming upper bound formulation, demonstrating that its radial objective test functions cannot asymptotically exclude packings with the Torquato--Stillinger density scalings. The agreement between these alternative approaches provides new evidence that exceptionally dense disordered sphere packings may exist in high dimensions and strengthens the case for the Torquato--Stillinger conjectural lower bound.
Carlo Vanoni, A. Guo, Salvatore Torquato· 0 citations
We investigate metric dimension and the localization game for several families of directed analogues of strongly regular graphs and their generalizations, adapting a probabilistic method of Babai (1980) for bounding the size of resolving sets in undirected strongly regular graphs. We derive upper bounds on the localization number and metric dimension depending on the order of the graph and the maximum number of common out-neighbours for a pair of vertices. We consider normally regular digraphs, so-called"ordinary graphs", classes of Deza digraphs, divisible design digraphs, nearly doubly regular tournaments, and certain doubly regular team tournaments. In particular, for asymmetric normally regular digraphs on $n$ vertices, we show that these invariants are bounded above by $O(\sqrt{n} \log n)$, and improve this to $O(\log n)$ for a class of doubly regular team tournaments.