9 research outputs found
Singleton-Optimal LRCs and Perfect LRCs via Cyclic and Constacyclic Codes
Locally repairable codes (LRCs) have emerged as an important coding scheme in
distributed storage systems (DSSs) with relatively low repair cost by accessing
fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs
have been widely investigated. Optimal LRCs via cyclic and constacyclic codes
provide significant benefit of elegant algebraic structure and efficient
encoding procedure. In this paper, we continue to consider the constructions of
optimal LRCs via cyclic and constacyclic codes with long code length.
Specifically, we first obtain two classes of -ary cyclic Singleton-optimal
-LRCs with length when and is
even, and length when and , respectively. To the best of our knowledge, this is the first
construction of -ary cyclic Singleton-optimal LRCs with length and
minimum distance . On the other hand, an LRC acheiving the
Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic
codes, we construct two new families of -ary perfect LRCs with length
, minimum distance and locality
New constructions of optimal -LRCs via good polynomials
Locally repairable codes (LRCs) are a class of erasure codes that are widely
used in distributed storage systems, which allow for efficient recovery of data
in the case of node failures or data loss. In 2014, Tamo and Barg introduced
Reed-Solomon-like (RS-like) Singleton-optimal -LRCs based on
polynomial evaluation. These constructions rely on the existence of so-called
good polynomial that is constant on each of some pairwise disjoint subsets of
. In this paper, we extend the aforementioned constructions of
RS-like LRCs and proposed new constructions of -LRCs whose code
length can be larger. These new -LRCs are all distance-optimal,
namely, they attain an upper bound on the minimum distance, that will be
established in this paper. This bound is sharper than the Singleton-type bound
in some cases owing to the extra conditions, it coincides with the
Singleton-type bound for certain cases. Combing these constructions with known
explicit good polynomials of special forms, we can get various explicit
Singleton-optimal -LRCs with new parameters, whose code lengths are
all larger than that constructed by the RS-like -LRCs introduced by
Tamo and Barg. Note that the code length of classical RS codes and RS-like LRCs
are both bounded by the field size. We explicitly construct the
Singleton-optimal -LRCs with length for any positive
integers and . When is
proportional to , it is asymptotically longer than that constructed via
elliptic curves whose length is at most . Besides, it allows more
flexibility on the values of and
Some new constructions of optimal linear codes and alphabet-optimal -locally repairable codes
In distributed storage systems, locally repairable codes (LRCs) are designed
to reduce disk I/O and repair costs by enabling recovery of each code symbol
from a small number of other symbols. To handle multiple node failures,
-LRCs are introduced to enable local recovery in the event of up to
failed nodes. Constructing optimal -LRCs has been a
significant research topic over the past decade. In \cite{Luo2022}, Luo
\emph{et al.} proposed a construction of linear codes by using unions of some
projective subspaces within a projective space. Several new classes of Griesmer
codes and distance-optimal codes were constructed, and some of them were proved
to be alphabet-optimal -LRCs.
In this paper, we first modify the method of constructing linear codes in
\cite{Luo2022} by considering a more general situation of intersecting
projective subspaces. This modification enables us to construct good codes with
more flexible parameters. Additionally, we present the conditions for the
constructed linear codes to qualify as Griesmer codes or achieve distance
optimality. Next, we explore the locality of linear codes constructed by
eliminating elements from a complete projective space. The novelty of our work
lies in establishing the locality as , , or -locality,
in contrast to the previous literature that only considered -locality.
Moreover, by combining analysis of code parameters and the C-M like bound for
-LRCs, we construct some alphabet-optimal -LRCs which
may be either Griesmer codes or not Griesmer codes. Finally, we investigate the
availability and alphabet-optimality of -LRCs constructed from our
modified framework.Comment: 25 page
Bounds and Constructions of Singleton-Optimal Locally Repairable Codes with Small Localities
Constructions of optimal locally repairable codes (LRCs) achieving
Singleton-type bound have been exhaustively investigated in recent years. In
this paper, we consider new bounds and constructions of Singleton-optimal LRCs
with minmum distance , locality and minimum distance and
locality , respectively. Firstly, we establish equivalent connections
between the existence of these two families of LRCs and the existence of some
subsets of lines in the projective space with certain properties. Then, we
employ the line-point incidence matrix and Johnson bounds for constant weight
codes to derive new improved bounds on the code length, which are tighter than
known results. Finally, by using some techniques of finite field and finite
geometry, we give some new constructions of Singleton-optimal LRCs, which have
larger length than previous ones
The hull of two classical propagation rules and their applications
Propagation rules are of great help in constructing good linear codes. Both
Euclidean and Hermitian hulls of linear codes perform an important part in
coding theory. In this paper, we consider these two aspects together and
determine the dimensions of Euclidean and Hermitian hulls of two classical
propagation rules, namely, the direct sum construction and the
-construction. Some new criteria for resulting codes
derived from these two propagation rules being self-dual, self-orthogonal or
linear complement dual (LCD) codes are given. As applications, we construct
some linear codes with prescribed hull dimensions and many new binary, ternary
Euclidean formally self-dual (FSD) LCD codes, quaternary Hermitian FSD LCD
codes and good quaternary Hermitian LCD codes which are optimal or have best or
almost best known parameters according to Datebase at
. Moreover, our methods contributes positively to
improve the lower bounds on the minimum distance of known LCD codes.Comment: 16 pages, 5 table
Intersections of linear codes and related MDS codes with new Galois hulls
Let denote the group of all semilinear
isometries on , where is a prime power. In this
paper, we investigate general properties of linear codes associated with
duals for . We show that
the dimension of the intersection of two linear codes can be determined by
generator matrices of such codes and their duals. We also show that
the dimension of hull of a linear code can be determined by a
generator matrix of it or its dual. We give a characterization on
dual and hull of a matrix-product code. We also investigate
the intersection of a pair of matrix-product codes. We provide a necessary and
sufficient condition under which any codeword of a generalized Reed-Solomon
(GRS) code or an extended GRS code is contained in its dual. As an
application, we construct eleven families of -ary MDS codes with new
-Galois hulls satisfying , which are not covered by the
latest papers by Cao (IEEE Trans. Inf. Theory 67(12), 7964-7984, 2021) and by
Fang et al. (Cryptogr. Commun. 14(1), 145-159, 2022) when
New Explicit Good Linear Sum-Rank-Metric Codes
Sum-rank-metric codes have wide applications in universal error correction
and security in multishot network, space-time coding and construction of
partial-MDS codes for repair in distributed storage. Fundamental properties of
sum-rank-metric codes have been studied and some explicit or probabilistic
constructions of good sum-rank-metric codes have been proposed. In this paper
we propose three simple constructions of explicit linear sum-rank-metric codes.
In finite length regime, numerous good linear sum-rank-metric codes from our
construction are given. Most of them have better parameters than previous
constructed sum-rank-metric codes. For example a lot of small block size better
linear sum-rank-metric codes over of the matrix size
are constructed for . Asymptotically our constructed sum-rank-metric
codes are closing to the Gilbert-Varshamov-like bound on sum-rank-metric codes
for some parameters. Finally we construct a linear MSRD code over an arbitrary
finite field with various matrix sizes
satisfying , , for any
given minimum sum-rank distance. There is no restriction on the block lengths
and parameters of these linear MSRD codes from the sizes
of the fields .Comment: 32 pages, revised version, merged with arXiv:2206.0233