I do not think there is a numerical example of BV out there, but generating one should be straightforward (though laborious) if you trace one of the examples in OpenFHE - make sure to set the user parameter keySwitchTechnique to BV as follows. parameters.SetKeySwitchTechnique(KeySwitchTechnique::BV);
We could help you better if you could narrow down the scope of your question and ask about specific details of BV.
To me, the single-level-decomposition BV is just like the hybrid mode, but dnum is equal to k_l - the number of limbs in the input ciphertext to be key-switched.
Have you also checked this resource (Section 2.2) while researching key switching? It provides a more accessible overview of the topic that might help.
Another paper that could also be useful for explaining BV key switching: An Improved RNS Variant of the BFV Homomorphic Encryption Scheme Section 3.1 introduces BV key switching in Brakerski’s scheme for the simplest case (see Key Generation and Relinearization). Then it gradually explains a more complex version (leading to the RNS version at the end).