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.
翻译:寻找$\mathbb{F}_2^t$中最大Sidon集合的大小已经超过40年的研究兴趣。为了解决这个问题,我们回顾了无和Sidon集合和具有最小距离大于或等于5的线性码之间的一一对应关系。我们关于码的主要贡献是一种基于Johnson bound的新的最小距离为5的线性码不存在性结果的加强版本。这在Sidon集合方面提供了最大Sidon集合的一般上界的改进。此外,我们在维数6上表征了最大的Sidon集合,即无法通过添加元素而不失去Sidon属性的Sidon集合,并通过计算机计算给出了维数7和8的所有可能大小。