Optimal codes and arcs for the generalized Hamming weights

TL;DR

This work studies the $r$-th generalized Hamming weights $d_r(C)$ of linear codes through a geometric lens, linking generalized weights to multisets of points in projective spaces and to arc/minihyper structures. It provides a purely geometric proof that the generalized Griesmer bound holds and that Griesmer codes for the usual Hamming distance can attain the bound for all generalized weights. The authors tabulate exact values $m_q^{(r)}(k-1,w)$ for binary codes with $k\le7$ and ternary codes with $k\le5$, delivering extensive exact formulas, constructions (including Solomon–Stiffler-type examples), and ILP verifications, thereby enriching code-design tables and highlighting cases where Griesmer-type bounds are tight. Overall, the paper advances understanding of the parameter landscape for generalized weights, offering concrete data and constructive methods that impact additive and linear code design in binary and ternary settings.

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.

Theorems & Definitions (62)

• Definition 1
• Lemma 1
• Lemma 2
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• Theorem 5