Constructing Large Nearly Orthogonal Sets over Finite Fields

Core Concepts

For every prime p, there exists a constant δ > 0 such that for every field F of characteristic p and for all integers k ≥ 2 and d ≥ k, there exists a k-nearly orthogonal set of at least dδ·k/ log k vectors in Fd. The size of this set is optimal up to the log k term in the exponent.

Abstract

The content discusses the problem of determining the largest possible size of nearly orthogonal sets over finite fields.
Key highlights:

For a field F and integers d and k, a set A ⊆ Fd is called k-nearly orthogonal if its members are non-self-orthogonal and every k + 1 vectors of A include an orthogonal pair.
Let α(d, k, F) denote the largest possible size of a k-nearly orthogonal subset of Fd.
A simple upper bound on α(d, k, F) stems from Ramsey theory, giving α(d, k, F) ≤ O(dk).
Over the real field R, the problem of determining α(d, k, R) was studied extensively, with lower and upper bounds being rather far apart.
The study of nearly orthogonal sets over finite fields was proposed by Codenotti, Pudlák, and Resta, motivated by questions in circuit complexity.
In contrast to the real field, it was shown that for the binary field F2, there exists a constant δ > 0 such that α(d, 2, F2) ≥ d1+δ for infinitely many integers d.
The main results of the paper are:

For every prime p, there exists a constant δ = δ(p) > 0 such that for every field F of characteristic p and for all integers k ≥ 2 and d ≥ k, it holds that β(d, k, F) ≥ dδ·k/ log k, where β(d, k, F) denotes the largest possible size of a set A ⊆ Fd of non-self-orthogonal vectors such that for every two (not necessarily disjoint) sets A1, A2 ⊆ A of size k + 1 each, there exist vectors v1 ∈ A1 and v2 ∈ A2 with ⟨v1, v2⟩ = 0.
For every prime p and every integer ℓ ≥ 1, there exists a constant δ = δ(p, ℓ) > 0 such that for every field F of characteristic p and for all integers k ≥ 2 and d ≥ k ≥ ℓ, it holds that α(d, k, ℓ, F) ≥ dδ·k/ log k, where α(d, k, ℓ, F) denotes the largest possible size of a (k, ℓ)-nearly orthogonal subset of Fd.

The proofs rely on probabilistic and spectral arguments, as well as the hypergraph container method.

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Key Insights Distilled From

Larger Nearly Orthogonal Sets over Finite Fields

by Isha... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.01057.pdf

Deeper Inquiries

How can the known lower and upper bounds on α(d, k, R) for general values of d and k be further narrowed?

The known lower and upper bounds on α(d, k, R) for general values of d and k can be further narrowed by exploring more refined probabilistic and spectral techniques. One approach could involve refining the probabilistic arguments used in constructing nearly orthogonal sets to potentially increase the size of these sets. Additionally, leveraging advanced spectral methods to analyze the structure of these sets and optimize their properties could lead to tighter bounds. By combining these approaches and potentially introducing new mathematical tools, it may be possible to narrow the gap between the lower and upper bounds on α(d, k, R) for various values of d and k.

What are the implications of the results in this paper for applications in circuit complexity and information theory?

The results presented in this paper have significant implications for applications in circuit complexity and information theory.

Circuit Complexity: Nearly orthogonal sets play a crucial role in circuit complexity, particularly in the design and analysis of circuits. The construction of large nearly orthogonal sets over finite fields can lead to the development of more efficient and robust circuit designs. By optimizing the size and properties of these sets, it becomes possible to enhance the performance and reliability of circuits, reducing computational complexity and improving overall efficiency.

Information Theory: In information theory, nearly orthogonal sets are fundamental in various coding and communication schemes. The results of this paper, which provide optimal sizes of nearly orthogonal sets over finite fields, can be directly applied to coding theory. By utilizing these sets in coding schemes, it is possible to enhance error correction capabilities, increase data transmission efficiency, and improve overall information processing systems. The results offer insights into constructing sets of vectors with specific orthogonality properties, which can be leveraged in various information-theoretic applications.

Can the techniques developed in this paper be extended to study nearly orthogonal sets over other algebraic structures, such as non-commutative rings?

The techniques developed in this paper can potentially be extended to study nearly orthogonal sets over other algebraic structures, including non-commutative rings. While the specific results and methods may need adaptation to suit the properties of non-commutative rings, the underlying probabilistic and spectral arguments can still be valuable in analyzing nearly orthogonal sets in these contexts.
By modifying the construction and analysis techniques to accommodate the algebraic properties of non-commutative rings, it may be possible to establish bounds and properties of nearly orthogonal sets in these structures. This extension could open up new avenues for exploring orthogonality and related concepts in non-commutative algebraic settings, offering insights into the structure and properties of sets of vectors in such systems.