12,020 research outputs found
New extremal singly even self-dual codes of lengths and
For lengths and , we construct extremal singly even self-dual codes
with weight enumerators for which no extremal singly even self-dual codes were
previously known to exist. We also construct new inequivalent extremal
doubly even self-dual codes with covering radius meeting the
Delsarte bound.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1706.0169
Rewriting Codes for Joint Information Storage in Flash Memories
Memories whose storage cells transit irreversibly between
states have been common since the start of the data storage
technology. In recent years, flash memories have become a very
important family of such memories. A flash memory cell has q
states—state 0.1.....q-1 - and can only transit from a lower
state to a higher state before the expensive erasure operation takes
place. We study rewriting codes that enable the data stored in a
group of cells to be rewritten by only shifting the cells to higher
states. Since the considered state transitions are irreversible, the
number of rewrites is bounded. Our objective is to maximize the
number of times the data can be rewritten. We focus on the joint
storage of data in flash memories, and study two rewriting codes
for two different scenarios. The first code, called floating code, is for
the joint storage of multiple variables, where every rewrite changes
one variable. The second code, called buffer code, is for remembering
the most recent data in a data stream. Many of the codes
presented here are either optimal or asymptotically optimal. We
also present bounds to the performance of general codes. The results
show that rewriting codes can integrate a flash memory’s
rewriting capabilities for different variables to a high degree
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Quasi-Perfect and Distance-Optimal Codes Sum-Rank Codes
Constructions of distance-optimal codes and quasi-perfect codes are
challenging problems and have attracted many attentions. In this paper, we give
the following three results.
1) If and , an infinite family of
distance-optimal -ary cyclic sum-rank codes with the block length
, the matrix size , the cardinality
and the minimum sum-rank distance four is constructed.
2) Block length and the matrix size distance-optimal
sum-rank codes with the minimum sum-rank distance four and the Singleton defect
four are constructed. These sum-rank codes are close to the sphere packing
bound , the Singleton-like bound and have much larger block length
.
3) For given positive integers satisfying , an infinite family
of quasi-perfect sum-rank codes with the matrix size , and the
minimum sum-rank distance three is also constructed. Quasi-perfect binary
sum-rank codes with the minimum sum-rank distance four are also given.
Almost MSRD -ary codes with the block lengths up to are given. We
show that more distance-optimal binary sum-rank codes can be obtained from the
Plotkin sum.Comment: 19 pages, only quasi-perfect sum-rank codes were constructed. Almost
MSRD codes with the block lengths up to were include
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