64 research outputs found
On Coset Leader Graphs of LDPC Codes
Our main technical result is that, in the coset leader graph of a linear
binary code of block length n, the metric balls spanned by constant-weight
vectors grow exponentially slower than those in .
Following the approach of Friedman and Tillich (2006), we use this fact to
improve on the first linear programming bound on the rate of LDPC codes, as the
function of their minimal distance. This improvement, combined with the
techniques of Ben-Haim and Lytsin (2006), improves the rate vs distance bounds
for LDPC codes in a significant sub-range of relative distances
Equiangular lines via matrix projection
In 1973, Lemmens and Seidel posed the problem of determining the maximum
number of equiangular lines in with angle and
gave a partial answer in the regime . At the other
extreme where is at least exponential in , recent breakthroughs
have led to an almost complete resolution of this problem. In this paper, we
introduce a new method for obtaining upper bounds which unifies and improves
upon previous approaches, thereby bridging the gap between the aforementioned
regimes, as well as significantly extending or improving all previously known
bounds when . Our method is based on orthogonal
projection of matrices with respect to the Frobenius inner product and it also
yields the first extension of the Alon-Boppana theorem to dense graphs, with
equality for strongly regular graphs corresponding to
equiangular lines in . Applications of our method in the complex
setting will be discussed as well.Comment: 39 pages, LaTeX; added new and improved results, improved
presentatio
Expander Graphs and Coding Theory
Expander graphs are highly connected sparse graphs which lie at the interface of many diļ¬erent ļ¬elds of study. For example, they play important roles in prime sieves, cryptography, compressive sensing, metric embedding, and coding theory to name a few. This thesis focuses on the connections between sparse graphs and coding theory. It is a major challenge to explicitly construct sparse graphs with good expansion properties, for example Ramanujan graphs. Nevertheless, explicit constructions do exist, and in this thesis, we survey many of these constructions up to this point including a new construction which slightly improves on an earlier edge expansion bound. The edge expansion of a graph is crucial in applications, and it is well-known that computing the edge expansion of an arbitrary graph is NP-hard. We present a simple algo-rithm for approximating the edge expansion of a graph using linear programming techniques. While Andersen and Lang (2008) proved similar results, our analysis attacks the problem from a diļ¬erent vantage point and was discovered independently. The main contribution in the thesis is a new result in fast decoding for expander codes. Current algorithms in the literature can decode a constant fraction of errors in linear time but require that the underlying graphs have vertex expansion at least 1/2. We present a fast decoding algorithm that can decode a constant fraction of errors in linear time given any vertex expansion (even if it is much smaller than 1/2) by using a stronger local code, and the fraction of errors corrected almost doubles that of Viderman (2013)
A Complete Linear Programming Hierarchy for Linear Codes
A longstanding open problem in coding theory is to determine the best
(asymptotic) rate of binary codes with minimum constant
(relative) distance . An existential lower bound was given by Gilbert
and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece,
Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte's
linear programs. To date these results remain the best known lower and upper
bounds on with no improvement even for the important class of
linear codes. Asymptotically, these bounds differ by an exponential factor in
the blocklength.
In this work, we introduce a new hierarchy of linear programs (LPs) that
converges to the true size of an optimum linear binary
code (in fact, over any finite field) of a given blocklength and distance
.
This hierarchy has several notable features:
(i) It is a natural generalization of the Delsarte LPs used in the first MRRW
bound.
(ii) It is a hierarchy of linear programs rather than semi-definite programs
potentially making it more amenable to theoretical analysis.
(iii) It is complete in the sense that the optimum code size can be retrieved
from level .
(iv) It provides an answer in the form of a hierarchy (in larger dimensional
spaces) to the question of how to cut Delsarte's LP polytopes to approximate
the true size of linear codes.
We obtain our hierarchy by generalizing the Krawtchouk polynomials and
MacWilliams inequalities to a suitable "higher-order" version taking into
account interactions of words. Our method also generalizes to
translation schemes under mild assumptions.Comment: 58 page
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