Perfect codes in Doob graphs

@article{Krotov2016PerfectCI,
  title={Perfect codes in Doob graphs},
  author={Denis S. Krotov},
  journal={Designs, Codes and Cryptography},
  year={2016},
  volume={80},
  pages={91-102}
}
  • D. Krotov
  • Published 23 July 2014
  • Mathematics, Computer Science
  • Designs, Codes and Cryptography
We study $$1$$1-perfect codes in Doob graphs $$D(m,n)$$D(m,n). We show that such codes that are linear over the Galois ring $$\mathrm {GR}(4^2)$$GR(42) exist if and only if there exist integers $$\gamma \ge 0$$γ≥0 and $$\delta >0$$δ>0 such that $$n=(4^{\gamma +\delta }-1)/3$$n=(4γ+δ-1)/3 and $$m=(4^{\gamma +2\delta }-4^{\gamma +\delta })/6$$m=(4γ+2δ-4γ+δ)/6. We also prove necessary conditions on $$(m,n)$$(m,n) for $$1$$1-perfect codes that are linear over $$Z_4$$Z4 (we call such codes additive… 
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