  The traditional definition of Integrated Interleaved (II) codes generally
assumes that the component nested codes are either Reed-Solomon (RS) or
shortened Reed-Solomon codes. By taking general classes of codes, we present a
recursive construction of Extended Integrated Interleaved (EII) codes into
multiple layers, a problem that brought attention in literature for II codes.
The multiple layer approach allows for a hierarchical scheme where each layer
of the code provides for a different locality. In particular, we present the
erasure-correcting capability of the new codes and we show that they are
ideally suited as Locally Recoverable (LRC) codes due to their hierarchical
locality and the small finite field required by the construction. Properties of
the multiple layer EII codes, like their minimum distance and dimension, as
well as their erasure decoding algorithms, parity-check matrices and
performance analysis, are provided and illustrated with examples. Finally, we
will observe that the parity-check matrices of high layer EII codes have low
density.
