O defektu slučajnih matrik z vrednostmi v končnih obsegih : delo diplomskega seminarja

Novoselec, Miha (author), Raič, Martin (mentor)

V delu obravnavamo porazdelitev defekta slučajnih matrik z vrednostmi v končnih obsegih. Pri enakomerni porazdelitvi elementov zadostuje štetje ustreznih matrik in lahko opišemo limitno porazdelitev defekta s konkretnim izrazom. Pri neenakomerni porazdelitvi elementov temu ni tako, vendar pa nad $GF(2)$ in dodatnih danih pogojih limitna porazdelitev defekta zopet konvergira k izrazu, ki smo ga izpeljali za enakomerno porazdelitev elementov.

• Language: Slovenian
• Keywords: slučajne matrike, limitna porazdelitev defekta matrike, linearne odvisnosti, množice stolpcev z ničelno vsoto
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2025
• PID: 20.500.12556/RUL-172990
• UDC: 519.2
• COBISS.SI-ID: 248764163
• Publication date in RUL: 12.09.2025

Secondary language

• Language: English
• Title: On defect of random matrices with values in finite fields

Abstract

In this thesis, we study the distribution of the defect of random matrices with entries from finite fields. For the uniform distribution of entries, counting the corresponding matrices allows us to describe the limiting defect distribution with an explicit formula. For the non-uniform distribution of entries, this approach is not directly applicable. Nevertheless, over the field $GF(2)$ and under additional assumptions, the limiting defect distribution again converges to the expression obtained for the uniform distribution of entries.

• Keywords: random matrices, limit distribution of matrix defect, linear dependencies, zero-sum column sets