On mobile sets in the binary hypercube
Yuriy Vasil'ev (Sobolev Institute of Mathematics, Novosibirsk,, Russia), Sergey Avgustinovich (Sobolev Institute of Mathematics, Novosibirsk,, Russia), Denis Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia)
TL;DR
This paper investigates the properties of mobile sets in high-dimensional binary hypercubes, revealing the existence of large, indivisible mobile sets that extend the concept of 1-perfect codes.
Contribution
It introduces the concept of mobile sets in the binary hypercube and constructs large, non-decomposable mobile sets in (4k+3)-dimensional hypercubes, extending previous understanding.
Findings
Existence of large mobile sets of size 2*6^k in (4k+3)-dimensional hypercubes
Such mobile sets cannot be split into smaller mobile sets
These sets are not simple extensions of lower-dimensional mobile sets
Abstract
If two distance-3 codes have the same neighborhood, then each of them is called a mobile set. In the (4k+3)-dimensional binary hypercube, there exists a mobile set of cardinality 2*6^k that cannot be split into mobile sets of smaller cardinalities or represented as a natural extension of a mobile set in a hypercube of smaller dimension. Keywords: mobile set; 1-perfect code.
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Finite Group Theory Research