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## 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
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Primary Language en Articles Author: Pani SENEVİRATNE 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 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.

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