For $N$ integer, it is known (Gilbert-Varshamov) that there exists a subset $S$ of $\{0,1\}^N$ such that $|S|$ is still exponential in $N$ and such that two elements $x, x'$ in $S$ are at least distant $\Omega(N)$ with respect to the Hamming metric.

Does the following result also hold:

Let $N \in \mathbb{N}$ and consider $\{0,1\}^N$ as well as a vector $\mathbf{w} \in \mathbb{R}_+^N$. Define the weighted Hamming metric such that for $\mathbf{x},\mathbf{x}' \in \{0,1\}^N$ we have $d_{\mathbf{w}}(\mathbf{x},\mathbf{x}') = \sum_{i=1}^N \mathbf{w}(i) |\mathbf{x}(i) - \mathbf{x'}(i)|$. Does there exist a subset $S$ of $\{0,1\}^N$ such that

- $|S| \geq 2^{C_1 \cdot N}$
- $d_{\mathbf{w}}(\mathbf{x},\mathbf{x}') \geq C_2 \|\mathbf{w}\|_1 N$

where $C_1, C_2$ are universal constants. That is, the selected elements will have disagreeing coordinates preferably where the cost is the highest, such that always a constant proportion of the total weight is captured.