Abstract. In 1991, Roth introduced a natural generalization of rank-metric codes, namely, tensor codes. The latter are defined to be subspaces of [Formula: see text]-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 corresponding 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 binomial moments and the weight distribution of a code with respect to a family of anticodes and establish a bijection between these invariants. We use the binomial moments to define the concept of a binomial moment determined code, which is an extremal code in relation to an inequality arising from them. Finally, we give MacWilliams identities for binomial moments.

We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. These lead to two new fundamental parameters of binary error-correcting codes, both of which measure the probability that the maximum likelihood decoder fails. We then derive various bounds for the cardinality and weight distribution of a binary code in terms of these new parameters, giving examples of codes meeting the bounds with equality.