The extended binary quadratic residue code of length 42 holds a 3-design

TL;DR

This paper demonstrates that the codewords of weight 10 in a specific extended binary quadratic residue code form a 3-design, revealing a novel combinatorial structure not explained by traditional theorems.

Contribution

It establishes the existence of a 3-(42,10,18) design within the codewords of the extended binary quadratic residue code of length 42, with automorphism group isomorphic to PSL(2,41).

Findings

Codewords of weight 10 form a 3-design.

Automorphism group is PSL(2,41).

Existence not explained by classical theorems.

Abstract

The codewords of weight $10$ of the $[42,21,10]$ extended binary quadratic residue code are shown to hold a design of parameters $3-(42,10,18).$ Its automorphism group is isomorphic to $PSL(2,41)$. Its existence can be explained neither by a transitivity argument, nor by the Assmus-Mattson theorem.