Optimal codes and arcs for the generalized Hamming weights

TL;DR

This paper provides a geometric proof of the Griesmer bound for generalized Hamming weights, explores constructions attaining this bound, and determines parameters of optimal small binary and ternary codes.

Contribution

It offers a geometric proof of the Griesmer bound and identifies optimal code parameters for small dimensions in binary and ternary cases.

Findings

A geometric proof of the Griesmer bound for generalized weights

Construction methods attaining the bound for large minimum distances

Optimal code parameters for small binary and ternary codes

Abstract

This text contains some notes on the Griesmer bound. In particular, we give a geometric proof of the Griesmer bound for the generalized weights and show that a Solomon--Stiffler type construction attains it if the minimum distance is sufficiently large. We also determine the parameters of optimal binary codes for dimensions at most seven and the optimal ternary codes for dimensions at most five.