Nontrivial t-Designs over Finite Fields Exist for All t

2013/06/10 by Arman Fazeli, Shachar Lovett, Fazeli, Arman +3
Computer Science · Engineering · #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1306.2088

openalex publication_date 2013/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A t-(n,k,λ) design over \F_q is a collection of k-dimensional subspaces of \F_qⁿ, called blocks, such that each t-dimensional subspace of \F_qⁿ is contained in exactly λ blocks. Such t-designs over \F_q are the q-analogs of conventional combinatorial designs. Nontrivial t-(n,k,λ) designs over \F_q are currently known to exist only for t ≤ 3. Herein, we prove that simple (meaning, without repeated blocks) nontrivial t-(n,k,λ) designs over \F_q exist for all t and q, provided that k > 12t and n is sufficiently large. This may be regarded as a q-analog of the celebrated Teirlinck theorem for combinatorial designs.

Related

• Large Sets of t-Designs over Finite Fields
• q-Analogs of t-Wise Balanced Designs from Borel Subgroups
• Combinatorial t-designs from quadratic functions
• Probabilistic Existence of Large Sets of Designs
• The Non-Existence of Block-Transitive Subspace Designs