The extended 1-perfect trades in small hypercubes

TL;DR

This paper classifies primary extended 1-perfect trades and Steiner trades of specific lengths in small hypercubes using computer-aided methods, providing a comprehensive understanding of their structure and equivalence classes.

Contribution

It introduces a computer-aided classification of primary extended 1-perfect trades and Steiner trades of lengths 10 and 12, including all trades with parameters (5,6,12).

Findings

Classified all primary extended 1-perfect trades of length 10.

Classified constant-weight extended 1-perfect trades of length 12.

Classified all Steiner trades with parameters (5,6,12).

Abstract

An extended $1$-perfect trade is a pair $(T_0,T_1)$ of two disjoint binary distance-$4$ even-weight codes such that the set of words at distance $1$ from $T_0$ coincides with the set of words at distance $1$ from $T_1$. Such trade is called primary if any pair of proper subsets of $T_0$ and $T_1$ is not a trade. Using a computer-aided approach, we classify nonequivalent primary extended $1$-perfect trades of length $10$, constant-weight extended $1$-perfect trades of length $12$, and Steiner trades derived from them. In particular, all Steiner trades with parameters $(5,6,12)$ are classified.