We give a characterization for the binary linear constant weight codes by using the symmetric difference of the supports of the codewords. This characterization gives a correspondence between the set of binary linear constant weight codes and the set of partitions for the union of supports of the codewords. By using this correspondence, we present a formula for the order of the automorphism group of a binary linear constant weight code in terms of its parameters. This formula is a key step to determine more algebraic structures on constant weight codes with given parameters. Bonisoli [Bonisoli, A.: Every equidistant linear code is a sequence of dual Hamming codes. Ars Combinatoria 18, 181--186 (1984)] proves that the $q$-ary linear constant weight codes with the same parameters are equivalent (for the binary case permutation equivalent). We also give an alternative proof for Bonisoli's theorem by presenting an explicit permutation on symmetric difference of the supports of the codewords which gives the permutation equivalence between the binary linear constant weight codes.

翻译：本文利用码字支撑的对称差给出了二元线性等重码的一种刻画。这一刻画建立了二元线性等重码集合与码字支撑并集的分划之间的对应关系。借助这一对应，我们给出了一个以参数表示的二元线性等重码自同构群阶数的公式。该公式是确定给定参数下等重码更多代数结构的关键步骤。Bonisoli [Bonisoli, A.: Every equidistant linear code is a sequence of dual Hamming codes. Ars Combinatoria 18, 181--186 (1984)] 证明了具有相同参数的q元线性等重码是等价的（在二元情形下为置换等价）。我们还通过给出码字支撑对称差上的一个显式置换，为Bonisoli定理提供了另一种证明，该置换给出了二元线性等重码之间的置换等价。