Integral Matrices of Fixed Rank over Number Fields

2025/10/13 by Gargava, Nihar, Serban, Vlad, Viazovska, Maryna +1 · 1 citation
#11H06 #11H50 #68R01 #94B75 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.11673

Abstract

We prove an asymptotic formula for the number of fixed rank matrices with integer coefficients over a number field K/Q and bounded norm. As an application, we derive an approximate Rogers integral formula for discrete sets of module lattices obtained from lifts of algebraic codes. This in turn implies that the moment estimates of random lattices with a number field structure also carry through for large enough discrete sets of module lattices.

Citations

• Counting matrices over finite rank multiplicative groups
• On some matrix counting problems
• Moments of the number of points in a bounded set for number field lattices
• Dense packings via lifts of codes to division rings
• Random Ensembles of Lattices from Generalized Reductions
• Worst-case to average-case reductions for module lattices

Cited by

• Module lattices and their shortest vectors

Related

• Effective module lattices and their shortest vectors
• The rank of random matrices over finite fields
• Moments of the number of points in a bounded set for number field lattices
• On the rank of random matrices over finite fields
• On some matrix counting problems