Example of gapped spin chain with degenerate ground space

What are the examples of a one dimensional spin chain, with local interaction and degenerate ground space (degeneracy may be a function of n, such as log(n) etc, where n is the length of chain) and a gap to the higher energy states, where gap is independent of n?

Edit: I managed to find an example with ground state degeneracy scaling as $N+1$, where $N$ is length of chain. Hamiltonian is projector onto the state $| 01\rangle - q|10\rangle$. Gap depends on parameter q and is gapped for q<1 . Reference: arxiv.org/abs/cond-mat/9512120

The classical Ising model $$H=\sum_i \sigma^z_i\sigma^z_{i+1}$$ has two degenerate ground states with a constant gap above (2 for periodic boundary conditions). This can be easily generalized to models with more than 2 states and correspondingly higher ground space degeneracy (Potts models).
In the XXZ spin chain, there exists some regime where the ground state becomes doubly degenerated in the infinite volume limit. This degeneracy produces a gap in energy between the ground state and the first excited state. As far as I know, on the finite chain, there is no degeneracy at all : the gap is therefore not independent of $n$. I don't know if such system exists.