If you are referring to the recent published [FIPS 203](https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.203.ipd.pdf) and [FIPS 204](https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.204.ipd.pdf) standards which specify ML-KEM and ML-DSA respectively (both are Module Learning with Error primitives, which are a particular variety of LWE system), then the options are very limited.

1. No, $q=2^t$ is not a permitted parameterisation. In fact for ML-KEM the only permissible value is $q=3329$ and for ML-DSA the only permissible value is $q=8380417$. As such, there is no one-to-one relationship between $q$ and $n$. For ML-KEM $n$ can take the values 512, 768, and 1024 (note that ML-KEM uses the notation $n$ for the polynomial degree of ring elements for whereas the $n$ in Lee definitions more closely corresponds to $kn$ in the ML-KEM spefication. Likewise, in ML-DSA the permitted values of $n$ are 512, 640, and 896.

2. The permissible values for $m$ in ML-KEM are 512, 768, and 1024. The permissible values for $m$ in ML-DSA are 512, 768, and 1024. The permissible values correspond directly to the listed permissible values of $n$.

Note that in ML-KEM and ML-DSA, the $a_i$ are of a very structured form and that $s_i$ are drawn from an error distribution similar to the $e_i$. The exact specification of parameters is given in table 2 of FIPS 203 and table 1 of FIPS 204.