In 1991, Roth introduced a natural generalization of rank metric codes, namely tensor codes. The latter are defined to be subspaces of $r$-tensors where the ambient space is endowed with the tensor rank as a distance function. In this work, we describe the general class of tensor codes and we study their invariants that correspond to different families of anticodes. In our context, an anticode is a perfect space that has some additional properties. A perfect space is one that is spanned by tensors of rank 1. Our use of the anticode concept is motivated by an interest in capturing structural properties of tensor codes. In particular, we indentify four different classes of tensor anticodes and show how these gives different information on the codes they describe. We also define the generalized tensor binomial moments and the generalized tensor weight distribution of a code and establish a bijection between these invariants. We use the generalized tensor binomial moments to define the concept of an $i$-tensor BMD code, which is an extremal code in relation to an inequality arising from them. Finally, we give MacWilliams identities for generalized tensor binomial moments.
翻译:1991年,罗斯引入了一种自然的通用标准代码,即高频代码,后者的定义是,在环境空间具有高频级功能的分空间为$-十等的亚空间,环境空间具有高频级的远程功能。在这项工作中,我们描述高频代码的一般类别,并研究与反反代码不同类别相对应的异差代码。在我们这个背景下,反代码是一个拥有一些额外特性的完美空间。一个完美的空间是级级数的数万分之一。1 我们使用反代码概念的动机是有兴趣获取高频代码的结构属性。特别是,我们确定了四种不同的高压反代码,并展示了这些代码对所描述的代码的不同信息。我们还定义了普通的高压二元代码和一种代码的通用数重分布,并在这些变量之间设置了一个插点。我们使用普通的 高压二元代码来定义美元-十等 BMD 代码的概念,这是与它们产生的不平等性特征相关的一个极端代码。最后,我们给了 将 磁号 硬体 。