13,491 research outputs found
Optimal locally repairable codes of distance and via cyclic codes
Like classical block codes, a locally repairable code also obeys the
Singleton-type bound (we call a locally repairable code {\it optimal} if it
achieves the Singleton-type bound). In the breakthrough work of \cite{TB14},
several classes of optimal locally repairable codes were constructed via
subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in
\cite{TB14} are upper bounded by the code alphabet size . Recently, it was
proved through extension of construction in \cite{TB14} that length of -ary
optimal locally repairable codes can be in \cite{JMX17}. Surprisingly,
\cite{BHHMV16} presented a few examples of -ary optimal locally repairable
codes of small distance and locality with code length achieving roughly .
Very recently, it was further shown in \cite{LMX17} that there exist -ary
optimal locally repairable codes with length bigger than and distance
propositional to .
Thus, it becomes an interesting and challenging problem to construct new
families of -ary optimal locally repairable codes of length bigger than
.
In this paper, we construct a class of optimal locally repairable codes of
distance and with unbounded length (i.e., length of the codes is
independent of the code alphabet size). Our technique is through cyclic codes
with particular generator and parity-check polynomials that are carefully
chosen
Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex
Reed-Muller codes are among the most important classes of locally correctable
codes. Currently local decoding of Reed-Muller codes is based on decoding on
lines or quadratic curves to recover one single coordinate. To recover multiple
coordinates simultaneously, the naive way is to repeat the local decoding for
recovery of a single coordinate. This decoding algorithm might be more
expensive, i.e., require higher query complexity. In this paper, we focus on
Reed-Muller codes with usual parameter regime, namely, the total degree of
evaluation polynomials is , where is the code alphabet size
(in fact, can be as big as in our setting). By introducing a novel
variation of codex, i.e., interleaved codex (the concept of codex has been used
for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover
arbitrarily large number of coordinates of a Reed-Muller code
simultaneously at the cost of querying coordinates. It turns out that
our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that
accessing locations is in fact cheaper than repeating the procedure for
accessing a single location for times. Our estimation of success error
probability is based on error probability bound for -wise linearly
independent variables given in \cite{BR94}
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