272 research outputs found
Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Binary Codes
Given positive integers and , let denote the maximum size
of a binary code of length and minimum distance . The well-known
Gilbert-Varshamov bound asserts that , where
is the volume of a Hamming sphere of
radius . We show that, in fact, there exists a positive constant such
that whenever . The result follows by recasting the Gilbert- Varshamov bound into a
graph-theoretic framework and using the fact that the corresponding graph is
locally sparse. Generalizations and extensions of this result are briefly
discussed.Comment: 10 pages, 3 figures; to appear in the IEEE Transactions on
Information Theory, submitted August 12, 2003, revised March 28, 200
Asymptotic improvement of the Gilbert-Varshamov bound for linear codes
The Gilbert-Varshamov bound states that the maximum size A_2(n,d) of a binary
code of length n and minimum distance d satisfies A_2(n,d) >= 2^n/V(n,d-1)
where V(n,d) stands for the volume of a Hamming ball of radius d. Recently
Jiang and Vardy showed that for binary non-linear codes this bound can be
improved to A_2(n,d) >= cn2^n/V(n,d-1) for c a constant and d/n <= 0.499. In
this paper we show that certain asymptotic families of linear binary [n,n/2]
random double circulant codes satisfy the same improved Gilbert-Varshamov
bound.Comment: Submitted to IEEE Transactions on Information Theor
An Upper Bound on the Minimum Distance of LDPC Codes over GF(q)
In [1] a syndrome counting based upper bound on the minimum distance of
regular binary LDPC codes is given. In this paper we extend the bound to the
case of irregular and generalized LDPC codes over GF(q). The comparison to the
lower bound for LDPC codes over GF(q) and to the upper bound for non-binary
codes is done. The new bound is shown to lie under the Gilbert-Varshamov bound
at high rates.Comment: 4 pages, submitted to ISIT 201
Improving the Gilbert-Varshamov Bound by Graph Spectral Method
We improve Gilbert-Varshamov bound by graph spectral method. Gilbert graph
is a graph with all vectors in as vertices where
two vertices are adjacent if their Hamming distance is less than . In this
paper, we calculate the eigenvalues and eigenvectors of using the
properties of Cayley graph. The improved bound is associated with the minimum
eigenvalue of the graph. Finally we give an algorithm to calculate the bound
and linear codes which satisfy the bound
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
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