On the Grassmann space representing the lines of an affine space

AbstractIn 1982, Bichara and Mazzocca characterized the Grassmann space Gr(1,A) of the lines of an affine space A of dimension at least 3 over a skew-field K by means of the intersection properties of the three disjoint families Σ1,Σ2 and T of maximal singular subspaces of Gr(1,A). In this paper, we deal with the characterization of Gr(1,A) using only the family Σ=Σ1∪Σ2 of maximal singular subspaces

Automorphism groups of Grassmann codes

We use a theorem of Chow (1949) on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question of Beelen et al. (2010) concerning the permutation automorphism groups of affine Grassmann codes. Finally, we prove an analogue of Chow's theorem for the case of Schubert divisors in Grassmannians and then use it to determine the automorphism group of linear codes associated to such Schubert divisors. In the course of this work, we also give an alternative short proof of MacWilliams theorem concerning the equivalence of linear codes and a characterization of maximal linear subspaces of Schubert divisors in Grassmannians.Comment: revised versio

Minimum distance of Symplectic Grassmann codes

We introduce the Symplectic Grassmann codes as projective codes defined by symplectic Grassmannians, in analogy with the orthogonal Grassmann codes introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special class of Symplectic Grassmann codes. We describe the weight enumerator of the Lagrangian--Grassmannian codes of rank $2$ and $3$ and we determine the minimum distance of the line Symplectic Grassmann codes.Comment: Revised contents and biblograph