Some Extremal Self-Dual Codes and Unimodular Lattices in Dimension 40

TL;DR

This paper classifies extremal self-dual codes and unimodular lattices in dimension 40, revealing their structures and relationships, which advances understanding in coding theory and lattice theory.

Contribution

It provides the first comprehensive classification of extremal singly even self-dual codes and extremal odd unimodular lattices in dimension 40.

Findings

Classification of extremal singly even self-dual codes of length 40.

Classification of extremal odd unimodular lattices in dimension 40.

Identification of relationships between lattices and self-dual codes.

Abstract

In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through their relationships with extremal doubly even self-dual codes of length 40.