39,092 research outputs found
Semidefinite programming, binary codes and a graph coloring problem
Experts in information theory have long been interested in the maximal size, A(n, d), of a binary error-correcting code of length n and minimum distance d, The problem of determining A(n, d) involves both the construction of good codes and the search for good upper bounds. For quite some time now, Delsarte\u27s linear programming approach has been the dominant approach to obtaining the strongest general purpose upper bounds on the efficiency of error-correcting codes. From 1973 forward, the linear programming bound found many applications, but there were few significant theoretical advances until Schrijver proposed a new code upper bound via semidefinite programming in 2003. Using the Terwilliger algebra, a recently introduced extension of the Bose-Mesner algebra, Schrijver formulated a new SDP strengthening of the LP approach. In this project we look at the dual solutions of the semidefinite programming bound for binary error-correcting codes. We explore the combinatorial meaning of these variables for small n and d, such as n = 4 and d = 2. To obtain information like this, we wrote a computer program with both Matlab and CVX modules to get solution of our primal SDP formulation. Our program efficiently generates the primal solutions with corresponding constraints for any n and d. We also wrote a program in C++ to parse the output of the primal SDP problem, and another Matlab script to generate the dual SDP problem, which could be used in assigning combinatorial meaning to the values given in the dual optimal solution. Our code not only computes both the primal and dual optimal variable values, but allows the researcher to display them in meaningful ways and to explore their relationship and dependence on arameters. These values are expected to be useful for later study of the combinatorial meaning of such solutions
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
Combinatorial Alphabet-Dependent Bounds for Locally Recoverable Codes
Locally recoverable (LRC) codes have recently been a focus point of research
in coding theory due to their theoretical appeal and applications in
distributed storage systems. In an LRC code, any erased symbol of a codeword
can be recovered by accessing only a small number of other symbols. For LRC
codes over a small alphabet (such as binary), the optimal rate-distance
trade-off is unknown. We present several new combinatorial bounds on LRC codes
including the locality-aware sphere packing and Plotkin bounds. We also develop
an approach to linear programming (LP) bounds on LRC codes. The resulting LP
bound gives better estimates in examples than the other upper bounds known in
the literature. Further, we provide the tightest known upper bound on the rate
of linear LRC codes with a given relative distance, an improvement over the
previous best known bounds.Comment: To appear in IEEE Transactions on Information Theor
Linear programming bounds for codes in Grassmannian spaces
We introduce a linear programming method to obtain bounds on the cardinality
of codes in Grassmannian spaces for the chordal distance. We obtain explicit
bounds, and an asymptotic bound that improves on the Hamming bound. Our
approach generalizes the approach originally developed by P. Delsarte and
Kabatianski-Levenshtein for compact two-point homogeneous spaces.Comment: 35 pages, 1 figur
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