38,803 research outputs found
Large weight code words in projective space codes
AbstractRecently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points
Second-Order Weight Distributions
A fundamental property of codes, the second-order weight distribution, is
proposed to solve the problems such as computing second moments of weight
distributions of linear code ensembles. A series of results, parallel to those
for weight distributions, is established for second-order weight distributions.
In particular, an analogue of MacWilliams identities is proved. The
second-order weight distributions of regular LDPC code ensembles are then
computed. As easy consequences, the second moments of weight distributions of
regular LDPC code ensembles are obtained. Furthermore, the application of
second-order weight distributions in random coding approach is discussed. The
second-order weight distributions of the ensembles generated by a so-called
2-good random generator or parity-check matrix are computed, where a 2-good
random matrix is a kind of generalization of the uniformly distributed random
matrix over a finite filed and is very useful for solving problems that involve
pairwise or triple-wise properties of sequences. It is shown that the 2-good
property is reflected in the second-order weight distribution, which thus plays
a fundamental role in some well-known problems in coding theory and
combinatorics. An example of linear intersecting codes is finally provided to
illustrate this fact.Comment: 10 pages, accepted for publication in IEEE Transactions on
Information Theory, May 201
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
Equidistant Codes in the Grassmannian
Equidistant codes over vector spaces are considered. For -dimensional
subspaces over a large vector space the largest code is always a sunflower. We
present several simple constructions for such codes which might produce the
largest non-sunflower codes. A novel construction, based on the Pl\"{u}cker
embedding, for 1-intersecting codes of -dimensional subspaces over \F_q^n,
, where the code size is is
presented. Finally, we present a related construction which generates
equidistant constant rank codes with matrices of size
over \F_q, rank , and rank distance .Comment: 16 page
(2,1)-separating systems beyond the probabilistic bound
Building on previous results of Xing, we give new lower bounds on the rate of
intersecting codes over large alphabets. The proof is constructive, and uses
algebraic geometry, although nothing beyond the basic theory of linear systems
on curves. Then, using these new bounds within a concatenation argument, we
construct binary (2,1)-separating systems of asymptotic rate exceeding the one
given by the probabilistic method, which was the best lower bound available up
to now. This answers (negatively) the question of whether this probabilistic
bound was exact, which has remained open for more than 30 years. (By the way,
we also give a formulation of the separation property in terms of metric
convexity, which may be an inspirational source for new research problems.)Comment: Version 7 is a shortened version, so that numbering should match with
the journal version (to appear soon). Material on convexity and separation in
discrete and continuous spaces has been removed. Readers interested in this
material should consult version 6 instea
The maximum number of minimal codewords in an code
Upper and lower bounds are derived for the quantity in the title, which is
tabulated for modest values of and An application to graphs with many
cycles is given.Comment: 6 pp. Submitte
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