Duals of Affine Grassmann Codes and their Relatives

TL;DR

This paper explicitly determines the duals and minimum distances of affine Grassmann codes of any level, improves understanding of their automorphism groups, and shows they are generated by their minimum-weight codewords.

Contribution

It provides explicit duals, minimum distances, automorphism group analysis, and proves these codes are generated by their minimum-weight codewords, extending prior results.

Findings

Explicit duals for affine Grassmann codes of any level

Computed minimum distances of these codes

Proved codes are generated by minimum-weight codewords

Abstract

Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work [2]. Here we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results of [2] concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes.