### On the most Weight $w$ Vectors in a Dimension $k$ Binary Code

#### Abstract

Ahlswede, Aydinian, and Khachatrian 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.