# New Extremal binary self-dual codes from a Baumert-Hall array

## TL;DR

This paper introduces new methods for constructing extremal binary self-dual codes using Baumert-Hall arrays, resulting in numerous new codes with specific parameters and associated combinatorial designs.

## Contribution

It presents novel construction techniques for self-dual codes via Baumert-Hall arrays, producing many previously unknown extremal codes and related combinatorial designs.

## Key findings

- 46 new extremal binary self-dual codes of length 68
- 26 new best known Type II codes of length 72
- 8 new extremal Type II codes of length 80

## Abstract

In this work, we introduce new construction methods for self-dual codes using a Baumert-Hall array. We apply the constructions over the alphabets F_2 and F_4 + uF_4 and combine them with extension theorems and neighboring constructions. As a result, we construct 46 new extremal binary self-dual codes of length 68, 26 new best known Type II codes of length 72 and 8 new extremal Type II codes of length 80 that lead to new 3-(80,16,665) designs. Among the new codes of length 68 are the examples of codes with the rare \gamma= 5 parameter in W68;2. All these new codes are tabulated in the paper.