  Recently, Skabelund defined new maximal curves which are cyclic extensions of
the Suzuki and Ree curves. Previously, the now well-known GK curves were found
as cyclic extensions of the Hermitian curve. In this paper, we consider locally
recoverable codes constructed from these new curves, complementing that done
for the GK curve. Locally recoverable codes allow for the recovery of a single
symbol by accessing only a few others which form what is known as a recovery
set. If every symbol has at least two disjoint recovery sets, the code is said
to have availability. Three constructions are described, as each best fits a
particular situation. The first employs the original construction of locally
recoverable codes from curves by Tamo and Barg. The second yields codes with
availability by appealing to the use of fiber products as described by
Haymaker, Malmskog, and Matthews, while the third accomplishes availability by
taking products of codes themselves. We see that cyclic extensions of the
Deligne-Lusztig curves provide codes with smaller locality than those typically
found in the literature.
