Automorphism groups of Grassmann codes

TL;DR

This paper determines the automorphism groups of Grassmann codes and related codes, using geometric theorems, and provides new insights into their structure and symmetries, including an alternative proof of MacWilliams theorem.

Contribution

It extends Chow's theorem to Schubert divisors and analyzes automorphisms of Grassmann and affine Grassmann codes, answering open questions in the field.

Findings

Automorphism groups of Grassmann codes are explicitly characterized.

An analogue of Chow's theorem for Schubert divisors is proved.

An alternative proof of MacWilliams theorem is provided.

Abstract

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.