  Optimal locally repairable codes with information locality are considered.
Optimal codes are constructed, whose length is also order-optimal with respect
to a new bound on the code length derived in this paper. The length of the
constructed codes is super-linear in the alphabet size, which improves upon the
well known pyramid codes, whose length is only linear in the alphabet size. The
recoverable erasure patterns are also analyzed for the new codes. Based on the
recoverable erasure patterns, we construct generalized sector-disk (GSD) codes,
which can recover from disk erasures mixed with sector erasures in a more
general setting than known sector-disk (SD) codes. Additionally, the number of
sectors in the constructed GSD codes is super-linear in the alphabet size,
compared with known SD codes, whose number of sectors is only linear in the
alphabet size.
