Example of partial homomorphic encryption over $\mathbb{F}_{2^k}$? I'm looking for a partial additive homomorphic encryption over $\mathbb{F}_{2^k}$.

I know that, I can take an additive homomorphic encryption over $\mathbb{F}_2$ and just concatenate each bit that we encrypted.
(as suggested at :Additive homomorphic encryption over small fields)

But it seems very inefficiently, when  $k$ is not a constant.

All the other encryption schemes that I saw work over $Z_{pq}^*$ and some variants.

Thanks.
I think you are making a classical mistake: $\mathbb{Z}_{2^k}$ is not $\mathbb{F}_{2^k}$. Concatenating $k$ ciphertexts homomorphic over $\mathbb{Z}_{2}$ gives an encryption scheme homomorphic over $\mathbb{Z}_{2}^k \approx \mathbb{F}_{2^k}$, but this is not $\mathbb{Z}_{2^k}$ - intuitively, $\mathbb{Z}_{a} \times \mathbb{Z}_{b} \approx \mathbb{Z}_{ab}$ only when $a$ and $b$ are coprime.

So, if what you want is really an additively homomorphic encryption scheme over $\mathbb{Z}_{2^k}$, there is one: this paper describes an extension of the Goldwasser-Micali encryption scheme which does exactly this.

If, however, you need an encryption scheme over $\mathbb{F}_{2^k}$, then there is not really any alternatives to concatenating encryptions over $\mathbb{Z}_{2}$, except using an appropriate scheme based on LWE - but I'm not sure whether this would improve a lot over the "naive" solution, I do not have a good knowledge of lattice-based cryptography.