How to convert the nvidia-container-toolkit.list in Ubuntu

Here is the original nvidia-container-toolkit.list from GitHub:

https://nvidia.github.io/libnvidia-container/stable/deb/nvidia-container-toolkit.list

deb https://nvidia.github.io/libnvidia-container/stable/deb/$(ARCH) /
#deb https://nvidia.github.io/libnvidia-container/experimental/deb/$(ARCH) /

Here is the document from Nvidia:

https://docs.nvidia.com/datacenter/cloud-native/container-toolkit/latest/install-guide.html

curl -fsSL https://nvidia.github.io/libnvidia-container/gpgkey | sudo gpg --dearmor -o /usr/share/keyrings/nvidia-container-toolkit-keyring.gpg \
  && curl -s -L https://nvidia.github.io/libnvidia-container/stable/deb/nvidia-container-toolkit.list | \
    sed 's#deb https://#deb [signed-by=/usr/share/keyrings/nvidia-container-toolkit-keyring.gpg] https://#g' | \
    sudo tee /etc/apt/sources.list.d/nvidia-container-toolkit.list

I have tried to add the "signed-by" manually, and then replace the "$(ARCH)" with "amd64" or " amd64":

deb [signed-by=/usr/share/keyrings/nvidia.gpg] https://nvidia.github.io/libnvidia-container/stable/deb/amd64 /

or 

deb [signed-by=/usr/share/keyrings/nvidia.gpg] https://nvidia.github.io/libnvidia-container/stable/deb/ amd64 /

But the "apt-get update" failed with the following error:

E: Malformed line 1 in source list /etc/apt/sources.list.d/nvidia-container-toolkit.list (type)
E: The list of sources could not be read.

Here is the echo $(ARCH) in Ubuntu:

# echo $(ARCH)
ARCH: command not found

What's the correct form of "nvidia-container-toolkit.list"?

Update:

If I did not replace the $(ARCH), the error is still the same. I think the original nvidia document is wrong.

Update 2:

The problem may relate to the "$(ARCH)", what's $(ARCH) in the nvidia?