Year 2016, Volume 3 , Issue 1, Pages 37 - 44 2016-01-11

Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes

Pani SENEVİRATNE [1]


A generalized hypercube graph $\Q_n(S)$ has $\F_{2}^{n}=\{0,1\}^n$ as the vertex set and two vertices being adjacent whenever their mutual Hamming distance belongs to $S$, where $n \ge 1$ and $S\subseteq \{1,2,\ldots, n\}$. The graph $\Q_n(\{1\})$ is the $n$-cube, usually denoted by $\Q_n$. We study graph boolean products $G_1 = \Q_n(S)\times \Q_1, G_2 = \Q_{n}(S)\wedge \Q_1$, $G_3 = \Q_{n}(S)[\Q_1]$ and show that binary codes from neighborhood designs of $G_1, G_2$ and $G_3$ are self-orthogonal for all choices of $n$ and $S$. More over, we show that the class of codes $C_1$ are self-dual. Further we find subgroups of the automorphism group of these graphs and use these subgroups to obtain PD-sets for permutation decoding. As an example we find a full error-correcting PD set for the binary $[32, 16, 8]$ extremal self-dual code.

Graphs, Designs, Codes, Permutation decoding
  • A. Berrachedi, M. Mollard, On two problems about (0, 2)-graphs and interval-regular graphs, Ars
  • Combin. 49 (1998) 303–309.
  • W. Fish, J. D. Key, E. Mwambene, Graphs, designs and codes related to the n-cube, Discrete Math. 309(10) (2009) 3255–3269.
  • F. Harary, G. W. Wilcox, Boolean operations on graphs, Math. Scand. 20 (1967) 41–51.
  • W. C. Huffman. Codes and groups. In V. Pless and W. C. Huffman, Eds., Handbook of coding theory, Vol. 2, pp. 1345–1440, Elsevier Science Publishers, Amsterdam, The Netherlands, 1998.
  • J. D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph Qn, where n is even. In T. Shaska, W. C. Huffman, D. Joyner, and V. Ustimenko, Eds., Advances in Coding Theory and Cryptography, Series on Coding Theory and Cryptography, Vol. 3, pp. 152–159, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.
  • J. M. Laborde, R. M. Madani, Generalized hypercubes and (0, 2)-graphs, Discrete Math. 165/166 (1997) 447–459.
  • F. J. MacWilliams, N. J. A. Sloane, The theory of error-correcting codes, Amsterdam: North-Holland, 1998.
  • M. Mulder, (0, λ)-graphs and n-cubes, Discrete Math. 28(2) (1979) 179–188.
Primary Language en
Journal Section Articles
Authors

Author: Pani SENEVİRATNE

Dates

Publication Date : January 11, 2016

Bibtex @ { jacodesmath168462, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2016}, volume = {3}, pages = {37 - 44}, doi = {10.13069/jacodesmath.13099}, title = {Generalized hypercube graph \$\\Q\_n(S)\$, graph products and self-orthogonal codes}, key = {cite}, author = {Senevi̇ratne, Pani} }
APA Senevi̇ratne, P . (2016). Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes . Journal of Algebra Combinatorics Discrete Structures and Applications , 3 (1) , 37-44 . DOI: 10.13069/jacodesmath.13099
MLA Senevi̇ratne, P . "Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes" . Journal of Algebra Combinatorics Discrete Structures and Applications 3 (2016 ): 37-44 <https://dergipark.org.tr/en/pub/jacodesmath/issue/16093/168462>
Chicago Senevi̇ratne, P . "Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes". Journal of Algebra Combinatorics Discrete Structures and Applications 3 (2016 ): 37-44
RIS TY - JOUR T1 - Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes AU - Pani Senevi̇ratne Y1 - 2016 PY - 2016 N1 - doi: 10.13069/jacodesmath.13099 DO - 10.13069/jacodesmath.13099 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 37 EP - 44 VL - 3 IS - 1 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.13099 UR - https://doi.org/10.13069/jacodesmath.13099 Y2 - 2020 ER -
EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes %A Pani Senevi̇ratne %T Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes %D 2016 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 3 %N 1 %R doi: 10.13069/jacodesmath.13099 %U 10.13069/jacodesmath.13099
ISNAD Senevi̇ratne, Pani . "Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes". Journal of Algebra Combinatorics Discrete Structures and Applications 3 / 1 (January 2016): 37-44 . https://doi.org/10.13069/jacodesmath.13099
AMA Senevi̇ratne P . Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016; 3(1): 37-44.
Vancouver Senevi̇ratne P . Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016; 3(1): 37-44.