66,240 research outputs found
Lattices from Codes for Harnessing Interference: An Overview and Generalizations
In this paper, using compute-and-forward as an example, we provide an
overview of constructions of lattices from codes that possess the right
algebraic structures for harnessing interference. This includes Construction A,
Construction D, and Construction (previously called product
construction) recently proposed by the authors. We then discuss two
generalizations where the first one is a general construction of lattices named
Construction subsuming the above three constructions as special cases
and the second one is to go beyond principal ideal domains and build lattices
over algebraic integers
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
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
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