The weight distribution of the self-dual $[128,64]$ polarity design code

TL;DR

This paper computes the weight distribution of a specific binary self-dual code derived from a polarity design in projective geometry and shows certain neighbor codes do not exist with specific minimum distances.

Contribution

It provides the explicit weight distribution of the self-dual [128,64] code from a polarity design and proves the non-existence of certain self-dual neighbors with given minimum distances.

Findings

Weight distribution of the [128,64] code is explicitly computed.

Proves no self-dual [128,64,d] neighbors with d=20 or 24 exist for the given codes.

Enhances understanding of the structure and limitations of self-dual codes from polarity designs.

Abstract

The weight distribution of the binary self-dual $[128,64]$ code being the extended code $C^{*}$ of the code $C$ spanned by the incidence vectors of the blocks of the polarity design in $PG(6,2)$ [11] is computed. It is shown also that $R(3,7)$ and $C^{*}$ have no self-dual $[128,64,d]$ neighbor with $d \in \{ 20, 24 \}$.