On the most weight \(w\) vectors in a dimension \(k\) binary code (Q612922)

Summary: \textit{R. Ahlswede}, \textit{H. Aydinian}, and \textit{L. Khachatrian} [``Maximum number of constant weight vertices of the unit \(n\)-cube contained in a \(k\)-dimensional subspace,'' Combinatorica 23, No.\,1, 5--22 (2003; Zbl 1046.05076)] posed the following problem: what is the maximum number of Hamming weight \(w\) vectors in a \(k\)-dimensional subspace of \(\mathbb F_2^n\)? The answer to this question could be relevant to coding theory, since it sheds light on the weight distributions of binary linear codes. We give some partial results. We also provide a conjecture for the complete solution when \(w\) is odd as well as for the case \(k\geq 2w\) and \(w\) even. One tool used to study this problem is a linear map that decreases the weight of nonzero vectors by a constant. We characterize such maps.