Rank properties of subspaces of symmetric and Hermitian matrices over finite fields

Jean-Guillaume Dumas (1, 2) , Rod Gow (2) , John Sheekey (2)

1 CASYS - Calculs Algébriques et Systèmes Dynamiques
2 CSI - Claude Shannon Institute

We investigate constant rank subspaces of symmetric and Hermitian matrices over finite fields, using a double counting method related to the number of common zeros of the corresponding subspaces of symmetric bilinear and Hermitian forms. We obtain optimal bounds for the dimensions of constant rank subspaces of Hermitian matrices, and good bounds for the dimensions of subspaces of symmetric and Hermitian matrices whose non-zero elements all have odd rank.

Mots clés: Matrix; Rank; Constant rank subspace; Symmetric matrix; Hermitian matrix; Common zeros

Domaines: Calcul formel [cs.SC]; Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Jean-Guillaume Dumas, Rod Gow, John Sheekey. Rank properties of subspaces of symmetric and Hermitian matrices over finite fields. Finite Fields and Their Applications, 2011, 17 (6), pp.504-520. ⟨10.1016/j.ffa.2011.03.001⟩. ⟨hal-00699691⟩