On the Maximum Probability of Full Rank of Random Matrices over Finite Fields

Marija Delić and Jelena Ivetić ()
Additional contact information
Marija Delić: Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
Jelena Ivetić: Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia

Mathematics, 2025, vol. 13, issue 3, 1-8

Abstract: The problem of determining the conditions under which a random rectangular matrix is of full rank is a fundamental question in random matrix theory, with significant implications for coding theory, cryptography, and combinatorics. In this paper, we study the probability of full rank for a K × N random matrix over the finite field F q , where q is a prime power, under the assumption that the rows of the matrix are sampled independently from a probability distribution P over F q N . We demonstrate that the probability of full rank attains a local maximum when the distribution P is uniform over F q N ∖ { 0 } , for any K ⩽ N and prime power q . Moreover, we establish that this local maximum is also a global maximum in the special case where K = 2 . These results highlight the optimality of the uniform distribution in maximizing full rank and represent a significant step toward solving the broader problem of maximizing the probability of full rank for random matrices over finite fields.

Keywords: random matrices; probability of full rank; linear independence; finite fields; uniform distribution; local maximum (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/3/540/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/3/540/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:3:p:540-:d:1585043

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().