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Some codes and designs invariant under the groups $S_7$ and $S_8$ | ||
| Journal of Mahani Mathematical Research | ||
| دوره 13، شماره 1 - شماره پیاپی 26، بهمن 2023، صفحه 511-524 اصل مقاله (504.76 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22103/jmmr.2023.21316.1430 | ||
| نویسنده | ||
| Reza Kahkeshani* | ||
| Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran | ||
| چکیده | ||
| We use the Key-Moori Method 1 and examine 1-designs and codes from the representations of the alternating group $A_7$. It is shown that a self-dual symmetric 2-$(35,18,9)$ design and an optimal even binary $[21,14,4]$ LCD code are found such that they are invariant under the full automorphism groups $S_8$ and $S_7$, respectively. Moreover, designs with parameters 1-$(21,l,k_{1,l})$ and 1-$(35,l,k_{2,l})$ are obtained, where $\omega$ is a codeword, $l=wt(\omega)$, $k_{1,l}=l|\omega^{S_7}|/21$ and $k_{2,l}=l|\omega^{S_7}|/35$. It is seen that there exist a 2-$(21,5,12)$ design with the full automorphism group $S_7$ among these 1-designs. | ||
| کلیدواژه‌ها | ||
| Code؛ Design؛ Automorphism group؛ Alternating group؛ Primitive permutation representation | ||
| مراجع | ||
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[13] Kahkeshani, R. (2020). On some designs constructed from the groups PSL2(q), q =53; 61; 64. Alg. Struc. Appl., 7(1), 59-67. https://doi.org/10.22034/AS.2020.1718
[14] Kahkeshani, R. (2022). Quaternary codes and a class of 2-designs invariant under the group A8. Alg. Struc. Appl., 9(1), 1-12. https://doi.org/10.22034/AS.2021.2254
[15] Kahkeshani, R. (2022). 1-Designs Constructed from the Groups PSL2(81) and PSL2(89). J. Mahani Math. Res., 11(2), 1-8. https://doi.org/10.22103/JMMRC.2022.18186.1166
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[22] Moori, J., & Saeidi, A. (2018). Constructing some designs invariant under PSL2(q), q even. Commun. Alg., 46(1), 160-166. https://doi.org/10.1080/00927872.2017.1316854 | ||
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