
TL;DR
This paper derives explicit formulas for Gauss sums over certain matrix groups over finite fields, using averaging techniques, and applies these results to matrix counting problems and bounds in additive combinatorics.
Contribution
It provides new explicit expressions for Gauss sums over linear groups and applies these to matrix enumeration and bound improvements.
Findings
Explicit formulas for Gauss sums over linear groups
Counting invertible matrices with zero-trace over finite fields
Improved bounds in additive combinatorics problems
Abstract
In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involves Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds by Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [ Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. (23) 2010, 501--513 ].
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Taxonomy
TopicsCoding theory and cryptography · Finite Group Theory Research · Limits and Structures in Graph Theory
