On perfect codes that do not contain Preparata-like codes

TL;DR

This paper proves that for lengths of the form 4^k-1, there are binary 1-perfect codes that do not contain any Preparata-like codes, expanding understanding of code structures.

Contribution

It demonstrates the existence of binary 1-perfect codes without Preparata-like codes for specific lengths, a novel result in coding theory.

Findings

Existence of such codes for all lengths 4^k-1

Construction methods for these codes

Implications for code classification and design

Abstract

We show that for every length of form $4^k-1$, there exists a binary $1$-perfect code that does not include any Preparata-like code.