On the distribution of the entries of a fixed-rank random matrix over a finite field

2023/07/26 by Carlo Sanna, Sanna, Carlo
Mathematics · #05A16 (Secondary) #11B33 #15B52 (Primary) 11T99 #Advanced Algebra and Geometry #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2307.14172

openalex publication_date 2023/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let r > 0 be an integer, let \mathbbF_q be a finite field of q elements, and let A be a nonempty proper subset of \mathbbF_q. Moreover, let M be a random m × n rank-r matrix over \mathbbF_q taken with uniform distribution. We prove, in a precise sense, that, as m, n → +∞ and r,q,A are fixed, the number of entries of M that belong to A approaches a normal distribution.

Related

• The central limit theorem for entries of random matrices with specific rank over finite fields
• A note on the rank of a sparse random matrix
• On the rank of random matrices over finite fields
• Rank of Near Uniform Matrices
• Universality of rational canonical form for random matrices over a finite field