# On transitive uniform partitions of F^n into binary Hamming codes

## TL;DR

This paper studies special partitions of binary vector spaces into Hamming code cosets, discovering new classes of highly symmetric partitions for large dimensions, which could impact coding theory and combinatorics.

## Contribution

It introduces and characterizes new classes of 2-transitive uniform partitions of F^n into Hamming code cosets for n=2^m-1, m>4, expanding understanding of such partitions.

## Key findings

- Found a class of nonequivalent 2-transitive uniform partitions
- Established existence for dimensions n=2^m-1, m>4
- Analyzed intersection properties of Hamming code cosets

## Abstract

We investigate transitive uniform partitions of the vector space $F^n$ of dimension $n$ over the Galois field $GF(2)$ into cosets of Hamming codes. A partition $P^n= \{H_0,H_1+e_1,\ldots,H_n+e_n\}$ of $F^n$ into cosets of Hamming codes $H_0,H_1,\ldots,H_n$ of length $n$ is said to be uniform if the intersection of any two codes $H_i$ and $H_j$, $i,j\in \{0,1,\ldots,n \}$ is constant, here $e_i$ is a binary vector in $F^n$ of weight $1$ with one in the $i$th coordinate position. For any $n=2^m-1$, $m>4$ we found a class of nonequivalent $2$-transitive uniform partitions of $F^n$ into cosets of Hamming codes.