Bounds on subspace codes based on totally isotropic subspace in symplectic spaces and extended symplectic spaces

In this paper, we propose the Sphere-packing bound, Singleton bound and Gilbert–Varshamov bound on the subspace codes [Formula: see text] based on totally isotropic subspaces in symplectic space [Formula: see text] and on the subspace codes [Formula: see text] based on totally isotropic subspace in extended symplectic space [Formula: see text].

SMALL ZEROS OF QUADRATIC FORMS OVER $\overline{\mathbb{Q}}$

Let N ≥ 2 be an integer, F a quadratic form in N variables over [Formula: see text], and [Formula: see text] an L-dimensional subspace, 1 ≤ L ≤ N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z,F). This provides an analogue over [Formula: see text] of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over [Formula: see text]. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over [Formula: see text]. This extends previous results of the author over number fields. All bounds on height are explicit.