15 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
Locally recoverable codes from automorphism groups of function fields of genus
A Locally Recoverable Code is a code such that the value of any single
coordinate of a codeword can be recovered from the values of a small subset of
other coordinates. When we have non overlapping subsets of cardinality
that can be used to recover the missing coordinate we say that a linear
code with length , dimension , minimum distance has
-locality and denote it by In this paper we provide a new upper bound for the minimum distance
of these codes. Working with a finite number of subgroups of cardinality
of the automorphism group of a function field of genus , we propose a construction of -codes and apply the results to some well known families
of function fields