ON THE MINIMAL COSET COVERINGS OF THE SET OF SINGULAR AND OF THE SET OF NONSINGULAR MATRICES

Abstract

It is determined minimum number of cosets over linear subspaces in $ \mathbb{F}_q $ necessary to cover following two sets of $ A (n \mathclose{\times} n) $ matrices. For one of the set of matrices $ \det{A} = 0 $ and for the other set $ \det{A} \neq 0 $. It is proved that for singular matrices this number is equal to $ 1 \mathclose{+} q \mathclose{+} q^2 \mathclose{+} \ldots \mathclose{+} q^{n-1} $ and for the nonsingular matrices it is equal to $ (q^n \mathclose{-} 1)(q^n \mathclose{-} q)(q^n \mathclose{-} q^2) \cdots (q^n \mathclose{-} q^{n-1}) / q^{\large{\binom{n}{2}}} $.