Automorphism groups of Grassmann codes

Automorphism groups of Grassmann codes


Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India
E-Mail: srg@math.iitb.ac.in

and

Krishna V. Kaipa

Department of Mathematics
Indian Institute of Science Education and Research Bhopal
Bhauri, Bhopal 462030, India
E-Mail: kaipa@iiserb.ac.in


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.


1 Introduction 1
2 Projective Systems and Automorphisms of Codes 4
3 Grassmann Codes 7
4 Affine Grassmann codes 12
5 Codes associated with the Schubert divisor of G(l,m) 17
References 25

This paper will be published in Finite Fields and Their Applications

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