# Binary codes and partial permutation decoding sets from the odd graphs

Open Mathematics (2014)

• Volume: 12, Issue: 9, page 1362-1371
• ISSN: 2391-5455

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## Abstract

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For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ωk, 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 $\overline{O\left(k\right)}$, is investigated.

## How to cite

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Washiela Fish, Roland Fray, and Eric Mwambene. "Binary codes and partial permutation decoding sets from the odd graphs." Open Mathematics 12.9 (2014): 1362-1371. <http://eudml.org/doc/269206>.

@article{WashielaFish2014,
abstract = {For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ωk, 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.},
author = {Washiela Fish, Roland Fray, Eric Mwambene},
journal = {Open Mathematics},
keywords = {Odd graphs; Binary codes; Automorphism group; Permutation decoding; odd graphs; binary codes; automorphism group; permutation decoding},
language = {eng},
number = {9},
pages = {1362-1371},
title = {Binary codes and partial permutation decoding sets from the odd graphs},
url = {http://eudml.org/doc/269206},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Washiela Fish
AU - Roland Fray
AU - Eric Mwambene
TI - Binary codes and partial permutation decoding sets from the odd graphs
JO - Open Mathematics
PY - 2014
VL - 12
IS - 9
SP - 1362
EP - 1371
AB - For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ωk, 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.
LA - eng
KW - Odd graphs; Binary codes; Automorphism group; Permutation decoding; odd graphs; binary codes; automorphism group; permutation decoding
UR - http://eudml.org/doc/269206
ER -

## References

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8. [8] Kroll H.-J., Vincenti R., PD-sets for the codes related to some classical varieties, Discrete Math., 2005, 301(1), 89–105 http://dx.doi.org/10.1016/j.disc.2004.11.020 Zbl1087.94024
9. [9] MacWilliams J., Permutation decoding of systematic codes, Bell System Tech. J., 1964, 43, 485–505 http://dx.doi.org/10.1002/j.1538-7305.1964.tb04075.x Zbl0116.35304
10. [10] MacWilliams F.J., Sloane N.J.A., The Theory of Error-Correcting Codes, North-Holland Math. Library, 16, North-Holland, Amsterdam, 1977
11. [11] Meredith G.H.J., Lloyd E.K., The footballers of Croam, J. Combinatorial Theory Ser. B, 1973, 15, 161–166 http://dx.doi.org/10.1016/0095-8956(73)90016-6 Zbl0248.05129

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