The Perfect Code

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

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Hamiltonicity of Minimum Distance Graphs of 1-Perfect Codes

The Electronic Journal of Combinatorics ◽

10.37236/2158 ◽

2012 ◽

Vol 19 (1) ◽

Cited By ~ 2

Author(s):

Alexander Mikhailovich Romanov

Keyword(s):

Minimum Distance ◽

Hamiltonian Cycle ◽

Distance Graph ◽

Perfect Code ◽

Perfect Codes ◽

Distance Graphs

A 1-perfect code $\mathcal{C}_{q}^{n}$ is called Hamiltonian if its minimum distance graph $G(\mathcal{C}_{q}^{n})$ contains a Hamiltonian cycle. In this paper, for all admissible lengths $n \geq 13$, we construct Hamiltonian nonlinear ternary 1-perfect codes, and for all admissible lengths $n \geq 21$, we construct Hamiltonian nonlinear quaternary 1-perfect codes. The existence of Hamiltonian nonlinear $q$-ary 1-perfect codes of length $N = qn + 1$ is reduced to the question of the existence of such codes of length $n$. Consequently, for $q = p^r$, where $p$ is prime, $r \geq 1$ there exist Hamiltonian nonlinear $q$-ary 1-perfect codes of length $n = (q ^{m} -1) / (q-1)$, $m \geq 2$. If $q =2, 3, 4$, then $ m \neq 2$. If $q =2$, then $ m \neq 3$.

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