Large spaces of bounded rank matrices revisited

Let n,p,r be positive integers with n≥p≥r. A rank-r‾ subset of n by p matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to r. A classical theorem of Flanders states that the dimension of a rank-r‾ linear subspace must be less than or equal to nr, and it characterizes the spaces with the critical dimension nr. Linear subspaces with dimension close to the critical one were later studied by Atkinson, Lloyd and Beasley over fields with large cardinality; their results were recently extended to all fields [18].Using a new method, we obtain a classification of rank-r‾ affine subspaces with large dimension, over all fields. This classification is then used to double the range of (large) dimensions for which the structure of rank-r‾ linear subspaces is known for all fields.