Fibonacci Designs

A Metis design is one for which v=r+k+1. This paper deals with Metis designs that are quasi-residual. The parameters of such designs and the corresponding symmetric designs can be expressed by Fibonacci numbers. Although the question of existence seems intractable because of the size of the designs, the nonexistence of corresponding difference sets can be dealt with in a substantive way. We also recall some inequalities for the number of fixed points of an automorphism of a symmetric design and suggest possible connections to the designs that would be the symmetric extensions of Metis designs.Comment: 7 page

P-Schemes and Deterministic Polynomial Factoring Over Finite Fields

We introduce a family of mathematical objects called P-schemes, where P is a poset of subgroups of a finite group G. A P-scheme is a collection of partitions of the right coset spaces H\G, indexed by H∈P, that satisfies a list of axioms. These objects generalize the classical notion of association schemes [BI84] as well as the notion of m-schemes [IKS09]. Based on P-schemes, we develop a unifying framework for the problem of deterministic factoring of univariate polynomials over finite field under the generalized Riemann hypothesis (GRH). More specifically, our results include the following: We show an equivalence between m-scheme as introduced in [IKS09] and P-schemes in the special setting that G is an multiply transitive permutation group and P is a poset of pointwise stabilizers, and therefore realize the theory of m-schemes as part of the richer theory of P-schemes. We give a generic deterministic algorithm that computes the factorization of the input polynomial ƒ(X) ∈ Fq[X] given a "lifted polynomial" ƒ~(X) of ƒ(X) and a collection F of "effectively constructible" subfields of the splitting field of ƒ~(X) over a certain base field. It is routine to compute ƒ~(X) from ƒ(X) by lifting the coefficients of ƒ(X) to a number ring. The algorithm then successfully factorizes ƒ(X) under GRH in time polynomial in the size of ƒ~(X) and F, provided that a certain condition concerning P-schemes is satisfied, for P being the poset of subgroups of the Galois group G of ƒ~(X) defined by F via the Galois correspondence. By considering various choices of G, P and verifying the condition, we are able to derive the main results of known (GRH-based) deterministic factoring algorithms [Hua91a; Hua91b; Ron88; Ron92; Evd92; Evd94; IKS09] from our generic algorithm in a uniform way. We investigate the schemes conjecture in [IKS09] and formulate analogous conjectures associated with various families of permutation groups, each of which has applications on deterministic polynomial factoring. Using a technique called induction of P-schemes, we establish reductions among these conjectures and show that they form a hierarchy of relaxations of the original schemes conjecture. We connect the complexity of deterministic polynomial factoring with the complexity of the Galois group G of ƒ~(X). Specifically, using techniques from permutation group theory, we obtain a (GRH-based) deterministic factoring algorithm whose running time is bounded in terms of the noncyclic composition factors of G. In particular, this algorithm runs in polynomial time if G is in Γk for some k=2O(√(log n), where Γk denotes the family of finite groups whose noncyclic composition factors are all isomorphic of subgroups of the symmetric group of degree k. Previously, polynomial-time algorithms for Γk were known only for bounded k. We discuss various aspects of the theory of P-schemes, including techniques of constructing new P-schemes from old ones, P-schemes for symmetric groups and linear groups, orbit P-schemes, etc. For the closely related theory of m-schemes, we provide explicit constructions of strongly antisymmetric homogeneous m-schemes for m≤3. We also show that all antisymmetric homogeneous orbit 3-schemes have a matching for m≥3, improving a result in [IKS09] that confirms the same statement for m≥4. In summary, our framework reduces the algorithmic problem of deterministic polynomial factoring over finite fields to a combinatorial problem concerning P-schemes, allowing us to not only recover most of the known results but also discover new ones. We believe progress in understanding P-schemes associated with various families of permutation groups will shed some light on the ultimate goal of solving deterministic polynomial factoring over finite fields in polynomial time.</p

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

On the Noncyclic Property of Sylvester Hadamard Matrices

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