About Some Large Cap Sets

Chapter 2 - Projective and Affine Geometry over Division Rings

Targets are matroids that arise from a nested sequence of flats in a projective geometry. This class of matroids was introduced by Nelson and Nomoto, who found the forbidden induced restrictions for binary targets. This paper generalizes their result to targets arising from projective geometries over $GF(q)$. We also consider targets arising from nested sequences of affine flats and determine the forbidden induced restrictions for affine targets.

Lattice-Theoretic Approach to Projective and Affine Geometry

In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g 1,g 2, …, g n , that are the product of linear factors. We then prove a punctured version which states, for simple zeros, that if f vanishes at nearly all, but not all, of the common zeros of g 1(X 1), …,g n (X n ) then every residue of f modulo the ideal generated by g 1, …, g n , has a large degree. This punctured Nullstellensatz is used to prove a blocking theorem for projective and affine geometries over an arbitrary field. This theorem has as corollaries a theorem of Alon and Furedi which gives a lower bound on the number of hyperplanes needed to cover all but one of the points of a hypercube and theorems of Bruen, Jamison and Brouwer and Schrijver which provides lower bounds on the number of points needed to block the hyperplanes of an affine space over a finite field.

A new definition for the dimension of a combinatorial t- (v,k,\lambda) design over a finite field is proposed. The complementary designs of the hyperplanes in a finite projective or affine geometry, and the finite Desarguesian planes in particular, are characterized as the unique (up to isomorphism) designs with the given parameters and minimum dimension. This generalizes a well-known characterization of the binary hyperplane designs in terms of their minimum 2-rank. The proof utilizes the q-ary analogue of the Hamming code, and a group-theoretic characterization of the classical designs.

Large language models (LLMs) have demonstrated tremendous capabilities in solving complex tasks, from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations), which can result in them making plausible but incorrect statements1,2. This hinders the use of current large models in scientific discovery. Here we introduce FunSearch (short for searching in the function space), an evolutionary procedure based on pairing a pretrained LLM with a systematic evaluator. We demonstrate the effectiveness of this approach to surpass the best-known results in important problems, pushing the boundary of existing LLM-based approaches3. Applying FunSearch to a central problem in extremal combinatorics—the cap set problem—we discover new constructions of large cap sets going beyond the best-known ones, both in finite dimensional and asymptotic cases. This shows that it is possible to make discoveries for established open problems using LLMs. We showcase the generality of FunSearch by applying it to an algorithmic problem, online bin packing, finding new heuristics that improve on widely used baselines. In contrast to most computer search approaches, FunSearch searches for programs that describe how to solve a problem, rather than what the solution is. Beyond being an effective and scalable strategy, discovered programs tend to be more interpretable than raw solutions, enabling feedback loops between domain experts and FunSearch, and the deployment of such programs in real-world applications.

Structural Properties of Models of N1-Categorical Theories

Cayley algebra and bracket algebra are important approaches to invariant computing in projective and affine geometries, but there are some difficulties in doing algebraic computation. In this paper we show how the principle “breefs” – bracket-oriented representation, elimination and expansion for factored and shortest results, can significantly simplify algebraic computations. We present several typical examples on automated theorem proving in conics and make detailed discussions on the procedure of applying the principle to automated geometric theorem proving.

Accurate regulation and measurement of a load parameters can be difficult when there are significant voltage drops between the power supply and the load. In particular, some methods of the load voltage management or regulation are known. These methods have these or those disadvantages. A new power supply method is offered for the remote load. This method is based on correction of the power supply output voltage by using an invariant relationship of the line. The affine ratio of three points, known in projective and affine geometry, is such invariant. Therefore, three samples of the output voltage and the corresponding samples of the input current are used. The affine ratio value does not depend from the line parameters and the measurement accuracy.

Constructing a Class of Symmetric Graphs

Three constructions of perfect authentication codes from projective geometry over finite fields

Metric geometry in the sense of Hjelmslev and Bachmann studies metric planes of a very general kind without any assumption about order, continuity and the existence and uniqueness of joining lines. An order structure can be defined in an additional step by introducing a relation of betweenness which satisfies the axioms of order of Hilbert’s Grundlagen der Geometrie, i.e., one-dimensional axioms which characterize the linear order of collinear points and a single plane order axiom which was proposed by Pasch. The Pasch axiom however is based on the assumption that any two points have a unique joining line. This is not necessarily satisfied by Cayley–Klein geometries (e.g. by Minkowskian planes) and even in plane absolute geometry the Pasch axiom is not a necessary condition for an ordering of the associated field of coordinates (see Sect. 5). The aim of this article is to introduce an order structure for the widest class of metric planes (without any assumption about the existence of joining lines, free mobility or some form of a parallel axiom) and to show that the correspondence between geometrical and algebraical order structures, which is well-known in affine and projective geometry, can be extended to plane absolute geometry. The article closes with a discussion of the role of the Pasch axiom in ordered metric geometry. An axiomatization of ordered metric planes in a first-order language is provided in an Appendix.

Let PG be the ‐dimensional projective space over and let be a simple graph of order for some . A design over is a collection of graphs (blocks) isomorphic to with the following properties: the vertex set of every block is a subspace of PG; every two distinct points of PG are adjacent in exactly blocks. This new definition covers, in particular, the well‐known concept of a design over corresponding to the case that is complete. In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of ‐decompositions over or for which is a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that is complete and , that is, the Steiner 2‐designs over a finite field. Also, we briefly touch the new topic of near resolvable designs over . This study has led us to some (probably new) collateral problems concerning difference sets. Supported by multiple examples, we conjecture the existence of infinite families of Γ‐decompositions over a finite field that can be obtained by suitably labeling the vertices of Γ with the elements of a Singer difference set.

Coded Caching is a promising solution to reduce the peak traffic in broadcast networks by prefetching the popular content close to end users and using coded transmissions. One of the chief issues of most coded caching schemes in literature is the issue of large subpacketization, i.e., they require each file to be divided into a large number of subfiles. In this work, we present a coded caching scheme using line graphs of bipartite graphs in conjunction with projective geometries over finite fields. The presented scheme achieves a rate Θ(K/log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> K) (K being the number of users, q is some prime power) with subexponential subpacketization q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O((logq K)^2)</sup> when cached fraction is upper bounded by a constant (M/N ≤ 1/q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> , for some positive integer α). Compared to earlier schemes, the presented scheme has a lower subpacketization (albeit possessing a higher rate). We also present a new subpacketization dependent lower bound on the rate for caching schemes in which each subfile is cached in the same number of users. Compared to the previously known bounds, this bound seems to perform better for a range of parameters of the caching system.

The axioms of constructive geometry