Sidon sets, sum-free sets and linear codes
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908742020235264 |
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| author | Czerwinski, Ingo Pott, Alexander |
| author_facet | Czerwinski, Ingo Pott, Alexander |
| contents | Finding the maximum size of a Sidon set in $\mathbb{F}_2^t$ is of research interest for more than 40 years. In order to tackle this problem we recall a one-to-one correspondence between sum-free Sidon sets and linear codes with minimum distance greater or equal 5. Our main contribution about codes is a new non-existence result for linear codes with minimum distance 5 based on a sharpening of the Johnson bound. This gives, on the Sidon set side, an improvement of the general upper bound for the maximum size of a Sidon set. Additionally, we characterise maximal Sidon sets, that are those Sidon sets which can not be extended by adding elements without loosing the Sidon property, up to dimension 6 and give all possible sizes for dimension 7 and 8 determined by computer calculations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07906 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sidon sets, sum-free sets and linear codes Czerwinski, Ingo Pott, Alexander Combinatorics Information Theory 11B13, 94B05, 94B65 Finding the maximum size of a Sidon set in $\mathbb{F}_2^t$ is of research interest for more than 40 years. In order to tackle this problem we recall a one-to-one correspondence between sum-free Sidon sets and linear codes with minimum distance greater or equal 5. Our main contribution about codes is a new non-existence result for linear codes with minimum distance 5 based on a sharpening of the Johnson bound. This gives, on the Sidon set side, an improvement of the general upper bound for the maximum size of a Sidon set. Additionally, we characterise maximal Sidon sets, that are those Sidon sets which can not be extended by adding elements without loosing the Sidon property, up to dimension 6 and give all possible sizes for dimension 7 and 8 determined by computer calculations. |
| title | Sidon sets, sum-free sets and linear codes |
| topic | Combinatorics Information Theory 11B13, 94B05, 94B65 |
| url | https://arxiv.org/abs/2304.07906 |