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1-Perfect Codes Over the Quad-Cube.
- Source :
- IEEE Transactions on Information Theory; Oct2022, Vol. 68 Issue 10, p6481-6504, 24p
- Publication Year :
- 2022
-
Abstract
- A vertex subset $S$ of a graph $G$ constitutes a 1-perfect code if the one-balls centered at the nodes in $S$ effect a vertex partition of $G$. This paper considers the quad-cube $CQ_{m}$ that is a connected $(m+2)$ -regular spanning subgraph of the hypercube $Q_{4m+2}$ , and shows that $CQ_{m}$ admits a vertex partition into 1-perfect codes iff $m=2^{k}-3$ , where $k\ge 2$. The scheme for that purpose makes use of a procedure by Jha and Slutzki that constructs Hamming codes using a Latin square. The result closely parallels the existence of a 1-perfect code over the dual-cube, which is another derivative of the hypercube. [ABSTRACT FROM AUTHOR]
- Subjects :
- HAMMING codes
GRAPH theory
DOMINATING set
SYMMETRIC matrices
HYPERCUBES
Subjects
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 68
- Issue :
- 10
- Database :
- Complementary Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 159210738
- Full Text :
- https://doi.org/10.1109/TIT.2022.3172924