  For an abelian group $H$ acting on the set $[\ell]$, an $(H,\ell)$-lift of a
graph $G_0$ is a graph obtained by replacing each vertex by $\ell$ copies, and
each edge by a matching corresponding to the action of an element of $H$.
  In this work, we show the following explicit constructions of expanders
obtained via abelian lifts. For every (transitive) abelian group $H \leqslant
\text{Sym}(\ell)$, constant degree $d \ge 3$ and $\epsilon > 0$, we construct
explicit $d$-regular expander graphs $G$ obtained from an $(H,\ell)$-lift of a
(suitable) base $n$-vertex expander $G_0$ with the following parameters:
  (i) $\lambda(G) \le 2\sqrt{d-1} + \epsilon$, for any lift size $\ell \le
2^{n^{\delta}}$ where $\delta=\delta(d,\epsilon)$,
  (ii) $\lambda(G) \le \epsilon \cdot d$, for any lift size $\ell \le
2^{n^{\delta_0}}$ for a fixed $\delta_0 > 0$, when $d \ge d_0(\epsilon)$, or
  (iii) $\lambda(G) \le \widetilde{O}(\sqrt{d})$, for lift size ``exactly''
$\ell = 2^{\Theta(n)}$.
  As corollaries, we obtain explicit quantum lifted product codes of Panteleev
and Kalachev of almost linear distance (and also in a wide range of parameters)
and explicit classical quasi-cyclic LDPC codes with wide range of circulant
sizes.
  Items $(i)$ and $(ii)$ above are obtained by extending the techniques of
Mohanty, O'Donnell and Paredes [STOC 2020] for $2$-lifts to much larger abelian
lift sizes (as a byproduct simplifying their construction). This is done by
providing a new encoding of special walks arising in the trace power method,
carefully "compressing'" depth-first search traversals. Result $(iii)$ is via a
simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of
polylog factors in the expansion.
