153 research outputs found
Low-density MDS codes and factors of complete graphs
We present a class of array code of size n×l, where l=2n or 2n+1, called B-Code. The distances of the B-Code and its dual are 3 and l-1, respectively. The B-Code and its dual are optimal in the sense that i) they are maximum-distance separable (MDS), ii) they have an optimal encoding property, i.e., the number of the parity bits that are affected by change of a single information bit is minimal, and iii) they have optimal length. Using a new graph description of the codes, we prove an equivalence relation between the construction of the B-Code (or its dual) and a combinatorial problem known as perfect one-factorization of complete graphs, thus obtaining constructions of two families of the B-Code and its dual, one of which is new. Efficient decoding algorithms are also given, both for erasure correcting and for error correcting. The existence of perfect one-factorizations for every complete graph with an even number of nodes is a 35 years long conjecture in graph theory. The construction of B-Codes of arbitrary odd length will provide an affirmative answer to the conjecture
Zigzag Codes: MDS Array Codes with Optimal Rebuilding
MDS array codes are widely used in storage systems to protect data against
erasures. We address the \emph{rebuilding ratio} problem, namely, in the case
of erasures, what is the fraction of the remaining information that needs to be
accessed in order to rebuild \emph{exactly} the lost information? It is clear
that when the number of erasures equals the maximum number of erasures that an
MDS code can correct then the rebuilding ratio is 1 (access all the remaining
information). However, the interesting and more practical case is when the
number of erasures is smaller than the erasure correcting capability of the
code. For example, consider an MDS code that can correct two erasures: What is
the smallest amount of information that one needs to access in order to correct
a single erasure? Previous work showed that the rebuilding ratio is bounded
between 1/2 and 3/4, however, the exact value was left as an open problem. In
this paper, we solve this open problem and prove that for the case of a single
erasure with a 2-erasure correcting code, the rebuilding ratio is 1/2. In
general, we construct a new family of -erasure correcting MDS array codes
that has optimal rebuilding ratio of in the case of erasures,
. Our array codes have efficient encoding and decoding
algorithms (for the case they use a finite field of size 3) and an
optimal update property.Comment: 23 pages, 5 figures, submitted to IEEE transactions on information
theor
Cyclic lowest density MDS array codes
Three new families of lowest density maximum-distance separable (MDS) array codes are constructed, which are cyclic or quasi-cyclic. In addition to their optimal redundancy (MDS) and optimal update complexity (lowest density), the symmetry offered by the new codes can be utilized for simplified implementation in storage applications. The proof of the code properties has an indirect structure: first MDS codes that are not cyclic are constructed, and then transformed to cyclic codes by a minimum-distance preserving transformation
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