5 research outputs found
Extended Integrated Interleaved Codes over any Field with Applications to Locally Recoverable Codes
Integrated Interleaved (II) and Extended Integrated Interleaved (EII) codes
are a versatile alternative for Locally Recoverable (LRC) codes, since they
require fields of relatively small size. II and EII codes are generally defined
over Reed-Solomon type of codes. A new comprehensive definition of EII codes is
presented, allowing for EII codes over any field, and in particular, over the
binary field . The traditional definition of II and EII codes is shown
to be a special case of the new definition. Improvements over previous
constructions of LRC codes, in particular, for binary codes, are given, as well
as cases meeting an upper bound on the minimum distance. Properties of the
codes are presented as well, in particular, an iterative decoding algorithm on
rows and columns generalizing the iterative decoding algorithm of product
codes. Two applications are also discussed: one is finding a systematic
encoding of EII codes such that the parity symbols have a balanced distribution
on rows, and the other is the problem of ordering the symbols of an EII code
such that the maximum length of a correctable burst is achieved.Comment: 25 page
Asymmetric Quantum Concatenated and Tensor Product Codes with Large Z-Distances
In this paper, we present a new construction of asymmetric quantum codes
(AQCs) by combining classical concatenated codes (CCs) with tensor product
codes (TPCs), called asymmetric quantum concatenated and tensor product codes
(AQCTPCs) which have the following three advantages. First, only the outer
codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes,
and any classical linear code can be used for the inner, which makes AQCTPCs
very easy to construct. Second, most AQCTPCs are highly degenerate, which means
they can correct many more errors than their classical TPC counterparts.
Consequently, we construct several families of AQCs with better parameters than
known results in the literature. Third, AQCTPCs can be efficiently decoded
although they are degenerate, provided that the inner and outer codes are
efficiently decodable. In particular, we significantly reduce the inner
decoding complexity of TPCs from to by
considering error degeneracy, where and are the block length of the
inner code and the outer code, respectively. Furthermore, we generalize our
concatenation scheme by using the generalized CCs and TPCs correspondingly.Comment: 36pages, accepted by IEEE Transactions on Communication
Universal and Dynamic Locally Repairable Codes with Maximal Recoverability via Sum-Rank Codes
Locally repairable codes (LRCs) are considered with equal or unequal
localities, local distances and local field sizes. An explicit two-layer
architecture with a sum-rank outer code is obtained, having disjoint local
groups and achieving maximal recoverability (MR) for all families of local
linear codes (MDS or not) simultaneously, up to a specified maximum locality . Furthermore, the local linear codes (thus the localities, local distances
and local fields) can be efficiently and dynamically modified without global
recoding or changes in architecture or outer code, while preserving the MR
property, easily adapting to new configurations in storage or new hot and cold
data. In addition, local groups and file components can be added, removed or
updated without global recoding. The construction requires global fields of
size roughly , for local groups and maximum or specified locality
. For equal localities, these global fields are smaller than those of
previous MR-LRCs when (global parities). For unequal localities,
they provide an exponential field size reduction on all previous best known
MR-LRCs. For bounded localities and a large number of local groups, the global
erasure-correction complexity of the given construction is comparable to that
of Tamo-Barg codes or Reed-Solomon codes with local replication, while local
repair is as efficient as for the Cartesian product of the local codes.
Reed-Solomon codes with local replication and Cartesian products are recovered
from the given construction when and , respectively. The given
construction can also be adapted to provide hierarchical MR-LRCs for all types
of hierarchies and parameters. Finally, subextension subcodes and sum-rank
alternant codes are introduced to obtain further exponential field size
reductions, at the expense of lower information rates
Multiple-Layer Integrated Interleaved Codes: A Class of Hierarchical Locally Recoverable Codes
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.Comment: 21 pages, 1 tabl
On Optimal Locally Repairable Codes and Generalized Sector-Disk Codes
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