In this paper we consider a finite-dimensional vector space \({\mathcal {P}}\) over the Galois field \({\text {GF}}(2),\) and the family \({\mathcal {B}}_k\) (respectively, \({\mathcal {B}}_k^*\)) of all the k-sets of elements of \(\mathcal {P}\) (respectively, of \({\mathcal {P}}^*= {\mathcal {P}} \setminus \{0\}\)) summing up to zero. We compute the parameters of the 3-design \(({\mathcal {P}},{\mathcal {B}}_k)\) for any (necessarily even) k, and of the 2-design \(({\mathcal {P}}^{*},{\mathcal {B}}_k^{*})\) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of \({\mathcal {P}}\), respectively of \({\mathcal {P}}^*\), that induce permutations of \({\mathcal {B}}_k\), respectively of \({\mathcal {B}}_k^*.\) In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight.

在本文中，我们考虑一个有限维向量空间\（{\ mathcal {P}} \）在所述伽罗瓦域\（{\文本{GF}}（2），\）和家庭\（{\ mathcal { B}} _ķ\） （分别为\（{\ mathcal {B}} K ^ * \ _）所有的）ķ的元素-Sets ）\（\ mathcal {P} \（分别的，\（{\ mathcal {P}} ^ * = {\ mathcal {P}} \ setminus \ {0 \} \））总和为零。我们针对任意k（必要为偶数）计算3设计\（（{{mathcal {P}}，{\ mathcal {B}} _ k）\）和2设计\（（{数学k {P}} ^ {*}，{\ mathcal {B}} _ k ^ {*}）\）任意k。同样，我们找到了二进制汉明码权重分布的新证明。此外，我们通过分别描述\（{\ mathcal {P}} ^ * \）的\（{\ mathcal {P}} \）的排列特征来找到上述设计的自同构群，它们引起\（{ {\ mathcal {B}} _ k \）分别为\（{\ mathcal {B}} _ k ^ *。\）特别是，这使人可以放宽对二进制汉明码和扩展的二进制汉明码作为排列组，仅保留给定汉明权重的码字。