Binary codes and partial permutation decoding sets from the odd graphs

• Autorzy: Washiela Fish , Roland Fray , Eric Mwambene
• Języki publikacji: EN

Abstrakty

EN

For 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.

Słowa kluczowe

EN

Odd graphs; Binary codes; Automorphism group; Permutation decoding

Kategorie tematyczne

• 05E18: Group actions on combinatorial structures
• 05C90: Applications
• 05B30: Other designs, configurations
• 94B05: Linear codes, general

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 9
• Strony: 1362-1371

Daty

wydano

2014-09-01

online

2014-05-08

Twórcy

autor

Washiela Fish

• University of the Western Cape

autor

Roland Fray

• University of the Western Cape

autor

Eric Mwambene

• University of the Western Cape

Bibliografia

• [1] Bailey R.F., Distance-Transitive Graphs, MMath thesis, University of Leeds, 2002
• [2] Balaban A.T., Chemical graphs, part XII: Combinatorial patterns, Rev. Roumaine Math. Pures Appl., 1972, 17, 3–16
• [3] Ball R.W., Maximal subgroups of symmetric groups, Trans. Amer. Math. Soc., 1966, 121(2), 393–407 http://dx.doi.org/10.1090/S0002-9947-1966-0202813-2
• [4] Biggs N., An edge-colouring problem, Amer. Math. Monthly, 1972, 79(9), 1018–1020 http://dx.doi.org/10.2307/2318076
• [5] Cameron P.J., Automorphism groups of graphs, In: Selected Topics in Graph Theory, 2, Academic Press, London, 1983, 89–127
• [6] Chen B.L., Lih K.-W., Hamiltonian uniform subset graphs, J. Combin. Theory Ser. B, 1987, 42(3), 257–263 http://dx.doi.org/10.1016/0095-8956(87)90044-X
• [7] Huffman W.C., Codes and groups, In: Handbook of Coding Theory, II, 2, North-Holland, Amsterdam, 1998, 1345–1440
• [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
• [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
• [10] MacWilliams F.J., Sloane N.J.A., The Theory of Error-Correcting Codes, North-Holland Math. Library, 16, North-Holland, Amsterdam, 1977
• [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

Identyfikatory

DOI

10.2478/s11533-014-0417-y