Differentical equations, dynamical systems and optimal control

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

F. I. Solov'eva

Sobolev Institute of Mathematics, pr. ac. Koptyuga 4, 630090, Novosibirsk, Russia

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.

Keywords: Hamming code, partition, uniform partition into Hamming codes, transitive partition, $2$-transitive partition, Reed–Muller code, dual code.

UDC: 519.72

MSC: 94B60

Received September 3, 2018, published June 21, 2019

Language: English

DOI: 10.33048/semi.2019.16.058

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