Unique representability and matroid reconstruction (Q1408269)

A matroid is deletion reconstructible (resp. \(k\)-reconstructible) if it is determined by its set of deletions (resp. \(k\)-element deletions). This article gives two methods for showing deletion reconstructibility of matroids that are uniquely GF\((q)\)-representable. The first one uses inclusion-exclusion and extends techniques in graph edge-reconstruction of Lovász. The second one extends techniques of Nash-Williams and is based on counting automorphisms. Bounds that are exponential in terms of rank are given for the number of points needed to ensure that a representable matroid is reconstructible, and quadratic bounds are given for binary and ternary matroids.