162 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
Families of locally separated Hamilton paths
We improve by an exponential factor the lower bound of K¨orner and Muzi for the cardinality of the largest family of Hamilton paths in a complete graph of n vertices in which the union of any two paths has maximum degree 4. The improvement is through an explicit construction while the previous bound was obtained by a greedy algorithm. We solve a similar problem for permutations up to an exponential factor
Families of locally separated Hamilton paths
We improve by an exponential factor the lower bound of K¨orner and Muzi for the cardinality of the largest family of Hamilton paths in a complete graph of n vertices in which the union of any two paths has maximum degree 4. The improvement is through an explicit construction while the previous bound was obtained by a greedy algorithm. We solve a similar problem for permutations up to an exponential factor
Upper Bounds on the Capacity of Binary Channels with Causal Adversaries
In this work we consider the communication of information in the presence of
a causal adversarial jammer. In the setting under study, a sender wishes to
communicate a message to a receiver by transmitting a codeword
bit-by-bit over a communication channel. The sender and the receiver do not
share common randomness. The adversarial jammer can view the transmitted bits
one at a time, and can change up to a -fraction of them. However, the
decisions of the jammer must be made in a causal manner. Namely, for each bit
the jammer's decision on whether to corrupt it or not must depend only on
for . This is in contrast to the "classical" adversarial
jamming situations in which the jammer has no knowledge of , or
knows completely. In this work, we present upper bounds (that
hold under both the average and maximal probability of error criteria) on the
capacity which hold for both deterministic and stochastic encoding schemes.Comment: To appear in the IEEE Transactions on Information Theory; shortened
version appeared at ISIT 201
On the VC-Dimension of Binary Codes
We investigate the asymptotic rates of length- binary codes with
VC-dimension at most and minimum distance at least . Two upper
bounds are obtained, one as a simple corollary of a result by Haussler and the
other via a shortening approach combining Sauer-Shelah lemma and the linear
programming bound. Two lower bounds are given using Gilbert-Varshamov type
arguments over constant-weight and Markov-type sets
Correcting Charge-Constrained Errors in the Rank-Modulation Scheme
We investigate error-correcting codes for a the
rank-modulation scheme with an application to flash memory
devices. In this scheme, a set of n cells stores information in the
permutation induced by the different charge levels of the individual
cells. The resulting scheme eliminates the need for discrete
cell levels, overcomes overshoot errors when programming cells (a
serious problem that reduces the writing speed), and mitigates the
problem of asymmetric errors. In this paper, we study the properties
of error-correcting codes for charge-constrained errors in the
rank-modulation scheme. In this error model the number of errors
corresponds to the minimal number of adjacent transpositions required
to change a given stored permutation to another erroneous
one—a distance measure known as Kendall’s τ-distance.We show
bounds on the size of such codes, and use metric-embedding techniques
to give constructions which translate a wealth of knowledge
of codes in the Lee metric to codes over permutations in Kendall’s
τ-metric. Specifically, the one-error-correcting codes we construct
are at least half the ball-packing upper bound
Spatially Coupled LDPC Codes Constructed from Protographs
In this paper, we construct protograph-based spatially coupled low-density
parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or
uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L,
we obtain a flexible family of code ensembles with varying rates and frame
lengths that can share the same encoding and decoding architecture for
arbitrary L. We demonstrate that the resulting codes combine the best features
of optimized irregular and regular codes in one design: capacity approaching
iterative belief propagation (BP) decoding thresholds and linear growth of
minimum distance with block length. In particular, we show that, for
sufficiently large L, the BP thresholds on both the binary erasure channel
(BEC) and the binary-input additive white Gaussian noise channel (AWGNC)
saturate to a particular value significantly better than the BP decoding
threshold and numerically indistinguishable from the optimal maximum
a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all
variable nodes in the coupled chain have degree greater than two,
asymptotically the error probability converges at least doubly exponentially
with decoding iterations and we obtain sequences of asymptotically good LDPC
codes with fast convergence rates and BP thresholds close to the Shannon limit.
Further, the gap to capacity decreases as the density of the graph increases,
opening up a new way to construct capacity achieving codes on memoryless
binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor
Constructions of q-Ary Constant-Weight Codes
This paper introduces a new combinatorial construction for q-ary
constant-weight codes which yields several families of optimal codes and
asymptotically optimal codes. The construction reveals intimate connection
between q-ary constant-weight codes and sets of pairwise disjoint combinatorial
designs of various types.Comment: 12 page
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