Self-dual codes and the non-existence of a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3

TL;DR

The paper proves the non-existence of a specific quasi-symmetric design by linking it to extremal self-dual codes and utilizing their classification.

Contribution

It establishes the non-existence of a certain quasi-symmetric 2-(37,9,8) design through coding theory methods and classification results.

Findings

The associated code must be contained in an extremal doubly even self-dual code of length 40.

Classification of extremal codes leads to the non-existence proof.

No such quasi-symmetric design exists.

Abstract

We prove that a certain binary linear code associated with the incidence matrix of a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3 must be contained in an extremal doubly even self-dual code of length 40. Using the classification of extremal doubly even self-dual codes of length 40, we show that a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3 does not exist.