Minimum distance of Line Orthogonal Grassmann Codes in even characteristic

TL;DR

This paper determines the minimum distance of orthogonal line-Grassmann codes over even characteristic fields and shows the equivalence of minimum weight codewords, also comparing with symplectic codes.

Contribution

It provides the minimum distance for orthogonal line-Grassmann codes in even characteristic and characterizes the minimum weight codewords, extending prior work to this case.

Findings

Minimum distance of orthogonal line-Grassmann codes for even q is established.

All minimum weight codewords are equivalent when q is even.

Symplectic line-Grassmann codes are proper subcodes of orthogonal codes, with codimension 2n.

Abstract

In this paper we determine the minimum distance of orthogonal line-Grassmann codes for $q$ even. The case $q$ odd was solved in "I. Cardinali, L. Giuzzi, K. Kaipa, A. Pasini, Line Polar Grassmann Codes of Orthogonal Type, J. Pure Applied Algebra." We also show that for $q$ even all minimum weight codewords are equivalent and that symplectic line-Grassmann codes are proper subcodes of codimension $2n$ of the orthogonal ones.