Binary Codes and Partial Permutation Decoding Sets from the Johnson Graphs

2014 ◽  
Vol 31 (5) ◽  
pp. 1381-1396 ◽  
Author(s):  
W. Fish
Keyword(s):  
Binary Codes ◽  
Permutation Decoding ◽  
Johnson Graphs

Codes from multipartite graphs and minimal permutation decoding sets

2015 ◽  
Vol 07 (04) ◽  
pp. 1550060
Author(s):  
P. Seneviratne
Keyword(s):  
Automorphism Group ◽  
Error Correcting Code ◽  
Binary Codes ◽  
Hard Problem ◽  
Complete Multipartite Graph ◽  
Decoding Algorithm ◽  
Permutation Decoding ◽  
Multipartite Graphs ◽  
Multipartite Graph ◽  
Complete Multipartite

Permutation decoding method developed by MacWilliams and described in [Permutation decoding of systematic codes, Bell Syst. Tech. J. 43 (1964) 485–505] is a decoding technique that uses a subset of the automorphism group of the code called a PD-set. The complexity of the permutation decoding algorithm depends on the size of the PD-set and finding a minimal PD-set for an error correcting code is a hard problem. In this paper we examine binary codes from the complete-multipartite graph [Formula: see text] and find PD-sets for all values of [Formula: see text] and [Formula: see text]. Further we show that these PD-sets are minimal when [Formula: see text] is odd and [Formula: see text].

scholarly journals Permutation decoding for binary codes from lattice graphs

2008 ◽  
Vol 308 (13) ◽  
pp. 2862-2867 ◽  
Author(s):  
J.D. Key ◽  
P. Seneviratne
Keyword(s):  
Binary Codes ◽  
Permutation Decoding ◽  
Lattice Graphs

Binary codes and partial permutation decoding sets from the odd graphs

2014 ◽  
Vol 12 (9) ◽  
Author(s):  
Washiela Fish ◽  
Roland Fray ◽  
Eric Mwambene
Keyword(s):  
Automorphism Group ◽  
Adjacency Matrix ◽  
Binary Code ◽  
Dual Code ◽  
Binary Codes ◽  
Permutation Decoding ◽  
Vertex Set ◽  
Information Set ◽  
The Relationship

AbstractFor k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ω{k}, the set of all k-subsets of Ω = {1, 2, …, 2k +1}, and any two of its vertices u and v constitute an edge [u, v] if and only if u ∩ v = /0. In this paper the binary code generated by the adjacency matrix of O(k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k 4 is determined. Lastly, the relationship between the dual code from O(k) and the code from its graph-theoretical complement $\overline {O(k)} $, is investigated.

scholarly journals Permutation decoding for the binary codes from triangular graphs

2004 ◽  
Vol 25 (1) ◽  
pp. 113-123 ◽  
Author(s):  
J.D. Key ◽  
J. Moori ◽  
B.G. Rodrigues
Keyword(s):  
Binary Codes ◽  
Permutation Decoding

Catalyst-Control over Sixfold Stereogenicity

2020 ◽  
Author(s):  
Xingxing Wu ◽  
Reto M. Witzig ◽  
Rodolphe Beaud ◽  
Christian Fischer ◽  
Daniel Häussinger ◽  
...  
Keyword(s):  
Conformational Analysis ◽  
Higher Order ◽  
Binary Codes ◽  
Rotational Barriers ◽  
Molecular Architectures

Governing higher-order stereogenicity is a long-standing goal in stereoselective catalysis, because it allows to achieve selectivity for more than a twofold number of stereoisomers per stereogenic unit. Current methods warrant control over the power of two stereoisomers and the configurations are routinely assigned using the descriptors ( R ) and ( S ), or related binary codes. In contrast, conformational analysis ranges beyond this dualistic treatment of stereoisomerism, which constitutes an unmet challenge for catalyst stereocontrolled processes. Herein, we now report that sixfold stereogenicity can be governed by stereoselective catalysis. By controlling a configurationally stable stereogenic axis with six large rotational barriers, a catalytic [2+2+2]-cyclotrimerization selectively governs the formation of one out of six stereoisomers with up to 0:0:2:98:0:0 stereocontrol. The underpinnings of conformational analysis and stereoselective catalysis are thereby conceptually reunited. Novel molecular architectures featuring distinct chemical topologies and unexplored chemical designs are anticipated from catalystcontrol over higher-order stereogenicities

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