89 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
New constructions of optimal -LRCs via good polynomials
Locally repairable codes (LRCs) are a class of erasure codes that are widely
used in distributed storage systems, which allow for efficient recovery of data
in the case of node failures or data loss. In 2014, Tamo and Barg introduced
Reed-Solomon-like (RS-like) Singleton-optimal -LRCs based on
polynomial evaluation. These constructions rely on the existence of so-called
good polynomial that is constant on each of some pairwise disjoint subsets of
. In this paper, we extend the aforementioned constructions of
RS-like LRCs and proposed new constructions of -LRCs whose code
length can be larger. These new -LRCs are all distance-optimal,
namely, they attain an upper bound on the minimum distance, that will be
established in this paper. This bound is sharper than the Singleton-type bound
in some cases owing to the extra conditions, it coincides with the
Singleton-type bound for certain cases. Combing these constructions with known
explicit good polynomials of special forms, we can get various explicit
Singleton-optimal -LRCs with new parameters, whose code lengths are
all larger than that constructed by the RS-like -LRCs introduced by
Tamo and Barg. Note that the code length of classical RS codes and RS-like LRCs
are both bounded by the field size. We explicitly construct the
Singleton-optimal -LRCs with length for any positive
integers and . When is
proportional to , it is asymptotically longer than that constructed via
elliptic curves whose length is at most . Besides, it allows more
flexibility on the values of and
How long can optimal locally repairable codes be?
A locally repairable code (LRC) with locality r allows for the recovery of any erased codeword symbol using only r other codeword symbols. A Singleton-type bound dictates the best possible trade-off between the dimension and distance of LRCs - an LRC attaining this trade-off is deemed optimal. Such optimal LRCs have been constructed over alphabets growing linearly in the block length. Unlike the classical Singleton bound, however, it was not known if such a linear growth in the alphabet size is necessary, or for that matter even if the alphabet needs to grow at all with the block length. Indeed, for small code distances 3,4, arbitrarily long optimal LRCs were known over fixed alphabets. Here, we prove that for distances d >=slant 5, the code length n of an optimal LRC over an alphabet of size q must be at most roughly O(d q^3). For the case d=5, our upper bound is O(q^2). We complement these bounds by showing the existence of optimal LRCs of length Omega_{d,r}(q^{1+1/floor[(d-3)/2]}) when d <=slant r+2. Our bounds match when d=5, pinning down n=Theta(q^2) as the asymptotically largest length of an optimal LRC for this case
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