Steiner systems $S(2,4,2^m)$ supported by a family of extended cyclic codes
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866929255075545088 |
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| author | Wang, Qi |
| author_facet | Wang, Qi |
| contents | In [C. Ding, An infinite family of Steiner systems $S(2,4,2^m)$ from cyclic codes, {\em J. Combin. Des.} 26 (2018), no.3, 126--144], Ding constructed a family of Steiner systems $S(2,4,2^m)$ for all $m \equiv 2 \pmod{4}$ from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems $S(2,4,2^m)$ for all $m \equiv 0 \pmod{4}$ supported by a family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system $S(2,4,2^m)$ for all even $m \geq 4$. This paper also determines the parameters of other $2$-designs supported by this family of extended cyclic codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_02310 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Steiner systems $S(2,4,2^m)$ supported by a family of extended cyclic codes Wang, Qi Combinatorics Information Theory 05B05, 11T71 In [C. Ding, An infinite family of Steiner systems $S(2,4,2^m)$ from cyclic codes, {\em J. Combin. Des.} 26 (2018), no.3, 126--144], Ding constructed a family of Steiner systems $S(2,4,2^m)$ for all $m \equiv 2 \pmod{4}$ from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems $S(2,4,2^m)$ for all $m \equiv 0 \pmod{4}$ supported by a family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system $S(2,4,2^m)$ for all even $m \geq 4$. This paper also determines the parameters of other $2$-designs supported by this family of extended cyclic codes. |
| title | Steiner systems $S(2,4,2^m)$ supported by a family of extended cyclic codes |
| topic | Combinatorics Information Theory 05B05, 11T71 |
| url | https://arxiv.org/abs/1904.02310 |