6,804 research outputs found
Covering -Symbol Metric Codes and the Generalized Singleton Bound
Symbol-pair codes were proposed for the application in high density storage
systems, where it is not possible to read individual symbols. Yaakobi, Bruck
and Siegel proved that the minimum pair-distance of binary linear cyclic codes
satisfies and introduced -symbol metric
codes in 2016. In this paper covering codes in -symbol metrics are
considered. Some examples are given to show that the Delsarte bound and the
Norse bound for covering codes in the Hamming metric are not true for covering
codes in the pair metric. We give the redundancy bound on covering radius of
linear codes in the -symbol metric and give some optimal codes attaining
this bound. Then we prove that there is no perfect linear symbol-pair code with
the minimum pair distance and there is no perfect -symbol metric code if
. Moreover a lot of cyclic and algebraic-geometric codes
are proved non-perfect in the -symbol metric. The covering radius of the
Reed-Solomon code in the -symbol metric is determined. As an application the
generalized Singleton bound on the sizes of list-decodable -symbol metric
codes is also presented. Then an upper bound on lengths of general MDS
symbol-pair codes is proved.Comment: 21 page
The Weights in MDS Codes
The weights in MDS codes of length n and dimension k over the finite field
GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with
parameters given by the MDS conjecture are shown to contain all k weights in
the range n-k+1 to n. The proof uses the covering radius of the dual codeComment: 5 pages, submitted to IEEE Trans. IT. This version 2 is the revised
version after the refereeing process. Accepted for publicatio
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
On the decoder error probability for Reed-Solomon codes
Upper bounds On the decoder error probability for Reed-Solomon codes are derived. By definition, "decoder error" occurs when the decoder finds a codeword other than the transitted codeword; this is in contrast to "decoder failure," which occurs when the decoder fails to find any codeword at all. These results imply, for example, that for a t error-correcting Reed-Solomon code of length q - 1 over GF(q), if more than t errors occur, the probability of decoder error is less than 1/t!
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