On the Assmus--Mattson type theorem for Type I and even formally self-dual codes

TL;DR

This paper extends the Assmus--Mattson theorem to near-extremal Type I and even formally self-dual codes, establishing the existence of certain combinatorial designs and proving the uniqueness of a specific design.

Contribution

It provides a new version of the Assmus--Mattson theorem applicable to these codes and demonstrates the uniqueness of a particular self-orthogonal design.

Findings

Existence of 1-designs and 2-designs for the codes.

Proof of the uniqueness of a self-orthogonal 2-(16,6,8) design.

Abstract

In the present paper, we give the Assmus--Mattson type theorem for near-extremal Type I and even formally self-dual codes. We show the existence of $1$-designs or $2$-designs for these codes. As a corollary, we prove the uniqueness of a self-orthogonal $2$-$(16,6,8)$ design.