359,361 research outputs found
General form of almost instantaneous fixed-to-variable-length codes
A general class of the almost instantaneous fixed-to-variable-length (AIFV)
codes is proposed, which contains every possible binary code we can make when
allowing finite bits of decoding delay. The contribution of the paper lies in
the following. (i) Introducing -bit-delay AIFV codes, constructed by
multiple code trees with higher flexibility than the conventional AIFV codes.
(ii) Proving that the proposed codes can represent any uniquely-encodable and
uniquely-decodable variable-to-variable length codes. (iii) Showing how to
express codes as multiple code trees with minimum decoding delay. (iv)
Formulating the constraints of decodability as the comparison of intervals in
the real number line. The theoretical results in this paper are expected to be
useful for further study on AIFV codes.Comment: submitted to IEEE Transactions on Information Theory. arXiv admin
note: text overlap with arXiv:1607.07247 by other author
Codes and Orbit Covers of Finite Abelian Groups
It is well known that the discrete analogue of a lattice is a linear code
which is a vector subspace of Hamming space . The set
is a finite field and . Our attempt is to
construct a class of lattices such that its discrete analogues are variable
length non-linear codes. Let and be two finite
groups, and let be a fixed set of generators for .
The homomorphism code is defined as the set of all homomorphisms from
to , denoted by, . To each homomorphism between and
, a codeword is associated, it is a vector of values
of on the generators in , that is, , where is the
image of , . We provide a design to
construct a variable length binary non-linear code called as automorphism orbit
code from a finite abelian -group of rank more than 1, where is a prime
number. For each finite abelian -group, the codewords of the automorphism
orbit code are variable length codewords called as automorphism orbit
codewords. Note that homomorphism codes are determined by homomorphisms between
groups, whereas automorphism orbit codes are specified by partitions of a
number, orbits of a group action, homomorphisms and automorphisms of groups. We
make use of elements of to present a cover
relation for bit strings of codewords of an automorphism orbit code and
formulate a lattice of variable length non-linear codes. Finally, we discuss
some information related to the future research work on connections to
representation theory of groups and algebras
Systematic Transmission With Fountain Parity Checks for Erasure Channels With Stop Feedback
In this paper, we present new achievability bounds on the maximal achievable
rate of variable-length stop-feedback (VLSF) codes operating over a binary
erasure channel (BEC) at a fixed message size . We provide new bounds
for VLSF codes with zero error, infinite decoding times and with nonzero error,
finite decoding times. Both new achievability bounds are proved by constructing
a new VLSF code that employs systematic transmission of the first bits
followed by random linear fountain parity bits decoded with a rank decoder. For
VLSF codes with infinite decoding times, our new bound outperforms the
state-of-the-art result for BEC by Devassy \emph{et al.} in 2016. We also give
a negative answer to the open question Devassy \emph{et al.} put forward on
whether the backoff to capacity at is fundamental. For VLSF
codes with finite decoding times, numerical evaluations show that the
achievable rate for VLSF codes with a moderate number of decoding times closely
approaches that for VLSF codes with infinite decoding times.Comment: 7 pages, double column, 4 figures; comments are welcome! changes in
v2: corrected 2 typos in v1. arXiv admin note: substantial text overlap with
arXiv:2205.1539
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
Synchronization recovery and state model reduction for soft decoding of variable length codes
Variable length codes exhibit de-synchronization problems when transmitted
over noisy channels. Trellis decoding techniques based on Maximum A Posteriori
(MAP) estimators are often used to minimize the error rate on the estimated
sequence. If the number of symbols and/or bits transmitted are known by the
decoder, termination constraints can be incorporated in the decoding process.
All the paths in the trellis which do not lead to a valid sequence length are
suppressed. This paper presents an analytic method to assess the expected error
resilience of a VLC when trellis decoding with a sequence length constraint is
used. The approach is based on the computation, for a given code, of the amount
of information brought by the constraint. It is then shown that this quantity
as well as the probability that the VLC decoder does not re-synchronize in a
strict sense, are not significantly altered by appropriate trellis states
aggregation. This proves that the performance obtained by running a
length-constrained Viterbi decoder on aggregated state models approaches the
one obtained with the bit/symbol trellis, with a significantly reduced
complexity. It is then shown that the complexity can be further decreased by
projecting the state model on two state models of reduced size
On Universal Properties of Capacity-Approaching LDPC Ensembles
This paper is focused on the derivation of some universal properties of
capacity-approaching low-density parity-check (LDPC) code ensembles whose
transmission takes place over memoryless binary-input output-symmetric (MBIOS)
channels. Properties of the degree distributions, graphical complexity and the
number of fundamental cycles in the bipartite graphs are considered via the
derivation of information-theoretic bounds. These bounds are expressed in terms
of the target block/ bit error probability and the gap (in rate) to capacity.
Most of the bounds are general for any decoding algorithm, and some others are
proved under belief propagation (BP) decoding. Proving these bounds under a
certain decoding algorithm, validates them automatically also under any
sub-optimal decoding algorithm. A proper modification of these bounds makes
them universal for the set of all MBIOS channels which exhibit a given
capacity. Bounds on the degree distributions and graphical complexity apply to
finite-length LDPC codes and to the asymptotic case of an infinite block
length. The bounds are compared with capacity-approaching LDPC code ensembles
under BP decoding, and they are shown to be informative and are easy to
calculate. Finally, some interesting open problems are considered.Comment: Published in the IEEE Trans. on Information Theory, vol. 55, no. 7,
pp. 2956 - 2990, July 200
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