“No perfect code”

2019 ◽  
pp. 33-67
Author(s):  
Barbara Barrow
Keyword(s):  
Perfect Code

scholarly journals Kernelization of edge perfect code and its variants

2016 ◽  
Vol 214 ◽  
pp. 145-152
Author(s):  
Minghui Jiang ◽  
Yong Zhang
Keyword(s):  
Perfect Code

The Perfect Code

2019 ◽  
pp. 63-77
Author(s):  
Terrence Holt
Keyword(s):  
Perfect Code

A binary perfect code of length 15 and codimension 0

1994 ◽  
Vol 4 (4) ◽  
pp. 213-220 ◽  
Author(s):  
Olof Heden
Keyword(s):  
Perfect Code ◽  
Binary Perfect Code

scholarly journals Embedding in a perfect code

2009 ◽  
Vol 17 (5) ◽  
pp. 419-423 ◽  
Author(s):  
Sergey V. Avgustinovich ◽  
Denis S. Krotov
Keyword(s):  
Perfect Code

scholarly journals On the size of the symmetry group of a perfect code

2011 ◽  
Vol 311 (17) ◽  
pp. 1879-1885 ◽  
Author(s):  
Olof Heden
Keyword(s):  
Symmetry Group ◽  
Perfect Code

scholarly journals Perfect Codes Over Induced Subgraphs of Unit Graphs of Ring of Integers Modulo n

2021 ◽  
Vol 20 ◽  
pp. 399-403
Author(s):  
Mohammad Hassan Mudaber ◽  
Nor Haniza Sarmin ◽  
Ibrahim Gambo
Keyword(s):  
Perfect Code ◽  
Perfect Codes ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Ring Of Integers ◽  
Unit Graph ◽  
Vertex Set

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

scholarly journals Hamiltonicity of Minimum Distance Graphs of 1-Perfect Codes

10.37236/2158 ◽  
2012 ◽  
Vol 19 (1) ◽  
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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