Let V be a finite dimensional vector space over a field. The collection of all subspaces of V of a fixed dimension can be viewed as a geometric object in the sense that it is a nice projective algebraic variety, known as Grassmann variety. When the base field is finite with q elements, the Grassmann variety can be viewed as the “moduli space” of q-ary linear (error correcting) codes of a fixed length and dimension. Some questions about codes can be profitably studied from this viewpoint. More interestingly, the Grassmann variety itself leads to a special class of linear codes that is arguably a good code. Variants of these obtained by considering generalizations of Grassmann varieties such as Schubert varieties or flag varieties, also lead to nice linear codes that seem worthy of investigation. We will outline these developments, while attempting to keep the prerequisites at a minimum.
