23,542 research outputs found
"Graph Entropy, Network Coding and Guessing games"
We introduce the (private) entropy of a directed graph (in a new network coding sense) as well as a number of related concepts. We show that the entropy of a directed graph is identical to its guessing number and can be bounded from below with the number of vertices minus the size of the graphâs shortest index code. We show that the Network Coding solvability of each speciïŹc multiple unicast network is completely determined by the entropy (as well as by the shortest index code) of the directed graph that occur by identifying each source node with each corresponding target node. Shannonâs information inequalities can be used to calculate up- per bounds on a graphâs entropy as well as calculating the size of the minimal index code. Recently, a number of new families of so-called non-shannon-type information inequalities have been discovered. It has been shown that there exist communication networks with a ca- pacity strictly ess than required for solvability, but where this fact cannot be derived using Shannonâs classical information inequalities. Based on this result we show that there exist graphs with an entropy that cannot be calculated using only Shannonâs classical information inequalities, and show that better estimate can be obtained by use of certain non-shannon-type information inequalities
Zero-error capacity of binary channels with memory
We begin a systematic study of the problem of the zero--error capacity of
noisy binary channels with memory and solve some of the non--trivial cases.Comment: 10 pages. This paper is the revised version of our previous paper
having the same title, published on ArXiV on February 3, 2014. We complete
Theorem 2 of the previous version by showing here that our previous
construction is asymptotically optimal. This proves that the isometric
triangles yield different capacities. The new manuscript differs from the old
one by the addition of one more pag
Non-linear index coding outperforming the linear optimum
The following source coding problem was introduced by Birk and Kol: a sender
holds a word , and wishes to broadcast a codeword to
receivers, . The receiver is interested in , and has
prior \emph{side information} comprising some subset of the bits. This
corresponds to a directed graph on vertices, where is an edge iff
knows the bit . An \emph{index code} for is an encoding scheme
which enables each to always reconstruct , given his side
information. The minimal word length of an index code was studied by
Bar-Yossef, Birk, Jayram and Kol (FOCS 2006). They introduced a graph
parameter, \minrk_2(G), which completely characterizes the length of an
optimal \emph{linear} index code for . The authors of BBJK showed that in
various cases linear codes attain the optimal word length, and conjectured that
linear index coding is in fact \emph{always} optimal.
In this work, we disprove the main conjecture of BBJK in the following strong
sense: for any and sufficiently large , there is an
-vertex graph so that every linear index code for requires codewords
of length at least , and yet a non-linear index code for
has a word length of . This is achieved by an explicit
construction, which extends Alon's variant of the celebrated Ramsey
construction of Frankl and Wilson.
In addition, we study optimal index codes in various, less restricted,
natural models, and prove several related properties of the graph parameter
\minrk(G).Comment: 16 pages; Preliminary version appeared in FOCS 200
Interlocked permutations
The zero-error capacity of channels with a countably infinite input alphabet
formally generalises Shannon's classical problem about the capacity of discrete
memoryless channels. We solve the problem for three particular channels. Our
results are purely combinatorial and in line with previous work of the third
author about permutation capacity.Comment: 8 page
Complexity and capacity bounds for quantum channels
We generalise some well-known graph parameters to operator systems by
considering their underlying quantum channels. In particular, we introduce the
quantum complexity as the dimension of the smallest co-domain Hilbert space a
quantum channel requires to realise a given operator system as its
non-commutative confusability graph. We describe quantum complexity as a
generalised minimum semidefinite rank and, in the case of a graph operator
system, as a quantum intersection number. The quantum complexity and a closely
related quantum version of orthogonal rank turn out to be upper bounds for the
Shannon zero-error capacity of a quantum channel, and we construct examples for
which these bounds beat the best previously known general upper bound for the
capacity of quantum channels, given by the quantum Lov\'asz theta number
The Zero-Undetected-Error Capacity Approaches the Sperner Capacity
Ahlswede, Cai, and Zhang proved that, in the noise-free limit, the
zero-undetected-error capacity is lower bounded by the Sperner capacity of the
channel graph, and they conjectured equality. Here we derive an upper bound
that proves the conjecture.Comment: 8 Pages; added a section on the definition of Sperner capacity;
accepted for publication in the IEEE Transactions on Information Theor
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