117 research outputs found
On the upper bound of the size of the r-cover-free families
Let T (r; n) denote the maximum number of subsets of an n-set satisfying the condition in the title. It is proved in a purely combinatorial way, that for n sufficiently large log 2 T (r; n) n 8 \Delta log 2 r r 2 holds. 1. Introduction The notion of the r-cover-free families was introduced by Kautz and Singleton in 1964 [17]. They initiated investigating binary codes with the property that the disjunction of any r (r 2) codewords are distinct (UD r codes). This led them to studying the binary codes with the property that none of the codewords is covered by the disjunction of r others (Superimposed codes, ZFD r codes; P. Erdos, P. Frankl and Z. Furedi called the correspondig set system r-cover-free in [7]). Since that many results have been proved about the maximum size of these codes. Various authors studied these problems basically from three different points of view, and these three lines of investigations were almost independent of each other. This is why many results were ..
List Decoding of Locally Repairable Codes
We show that locally repairable codes (LRCs) can be list decoded efficiently
beyond the Johnson radius for a large range of parameters by utilizing the
local error correction capabilities. The new decoding radius is derived and the
asymptotic behavior is analyzed. We give a general list decoding algorithm for
LRCs that achieves this radius along with an explicit realization for a class
of LRCs based on Reed-Solomon codes (Tamo-Barg LRCs). Further, a probabilistic
algorithm for unique decoding of low complexity is given and its success
probability analyzed
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