9,678 research outputs found
Query-Efficient Locally Decodable Codes of Subexponential Length
We develop the algebraic theory behind the constructions of Yekhanin (2008)
and Efremenko (2009), in an attempt to understand the ``algebraic niceness''
phenomenon in . We show that every integer ,
where , and are prime, possesses the same good algebraic property as
that allows savings in query complexity. We identify 50 numbers of this
form by computer search, which together with 511, are then applied to gain
improvements on query complexity via Itoh and Suzuki's composition method. More
precisely, we construct a -query LDC for every positive
integer and a -query
LDC for every integer , both of length , improving the
queries used by Efremenko (2009) and queries used by Itoh and
Suzuki (2010).
We also obtain new efficient private information retrieval (PIR) schemes from
the new query-efficient LDCs.Comment: to appear in Computational Complexit
Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping
In this paper, we provide for the first time a systematic comparison of
distribution matching (DM) and sphere shaping (SpSh) algorithms for short
blocklength probabilistic amplitude shaping. For asymptotically large
blocklengths, constant composition distribution matching (CCDM) is known to
generate the target capacity-achieving distribution. As the blocklength
decreases, however, the resulting rate loss diminishes the efficiency of CCDM.
We claim that for such short blocklengths and over the additive white Gaussian
channel (AWGN), the objective of shaping should be reformulated as obtaining
the most energy-efficient signal space for a given rate (rather than matching
distributions). In light of this interpretation, multiset-partition DM (MPDM),
enumerative sphere shaping (ESS) and shell mapping (SM), are reviewed as
energy-efficient shaping techniques. Numerical results show that MPDM and SpSh
have smaller rate losses than CCDM. SpSh--whose sole objective is to maximize
the energy efficiency--is shown to have the minimum rate loss amongst all. We
provide simulation results of the end-to-end decoding performance showing that
up to 1 dB improvement in power efficiency over uniform signaling can be
obtained with MPDM and SpSh at blocklengths around 200. Finally, we present a
discussion on the complexity of these algorithms from the perspective of
latency, storage and computations.Comment: 18 pages, 10 figure
Limits to Non-Malleability
There have been many successes in constructing explicit non-malleable codes for various classes of tampering functions in recent years, and strong existential results are also known. In this work we ask the following question:
When can we rule out the existence of a non-malleable code for a tampering class ??
First, we start with some classes where positive results are well-known, and show that when these classes are extended in a natural way, non-malleable codes are no longer possible. Specifically, we show that no non-malleable codes exist for any of the following tampering classes:
- Functions that change d/2 symbols, where d is the distance of the code;
- Functions where each input symbol affects only a single output symbol;
- Functions where each of the n output bits is a function of n-log n input bits.
Furthermore, we rule out constructions of non-malleable codes for certain classes ? via reductions to the assumption that a distributional problem is hard for ?, that make black-box use of the tampering functions in the proof. In particular, this yields concrete obstacles for the construction of efficient codes for NC, even assuming average-case variants of P ? NC
Subquadratic time encodable codes beating the Gilbert-Varshamov bound
We construct explicit algebraic geometry codes built from the
Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for
alphabet sizes at least 192. Messages are identied with functions in certain
Riemann-Roch spaces associated with divisors supported on multiple places.
Encoding amounts to evaluating these functions at degree one places. By
exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we
devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and
1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list)
decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent.
If \omega = 2, as widely believed, the encoding and decoding runtimes are
respectively nearly linear and nearly quadratic. Prior to this work, encoding
(resp. decoding) time of code families beating the Gilbert-Varshamov bound were
quadratic (resp. cubic) or worse
Optimal rate list decoding via derivative codes
The classical family of Reed-Solomon codes over a field \F_q
consist of the evaluations of polynomials f \in \F_q[X] of degree at
distinct field elements. In this work, we consider a closely related family
of codes, called (order ) {\em derivative codes} and defined over fields of
large characteristic, which consist of the evaluations of as well as its
first formal derivatives at distinct field elements. For large enough
, we show that these codes can be list-decoded in polynomial time from an
error fraction approaching , where is the rate of the code.
This gives an alternate construction to folded Reed-Solomon codes for achieving
the optimal trade-off between rate and list error-correction radius. Our
decoding algorithm is linear-algebraic, and involves solving a linear system to
interpolate a multivariate polynomial, and then solving another structured
linear system to retrieve the list of candidate polynomials . The algorithm
for derivative codes offers some advantages compared to a similar one for
folded Reed-Solomon codes in terms of efficient unique decoding in the presence
of side information.Comment: 11 page
Frame Permutation Quantization
Frame permutation quantization (FPQ) is a new vector quantization technique
using finite frames. In FPQ, a vector is encoded using a permutation source
code to quantize its frame expansion. This means that the encoding is a partial
ordering of the frame expansion coefficients. Compared to ordinary permutation
source coding, FPQ produces a greater number of possible quantization rates and
a higher maximum rate. Various representations for the partitions induced by
FPQ are presented, and reconstruction algorithms based on linear programming,
quadratic programming, and recursive orthogonal projection are derived.
Implementations of the linear and quadratic programming algorithms for uniform
and Gaussian sources show performance improvements over entropy-constrained
scalar quantization for certain combinations of vector dimension and coding
rate. Monte Carlo evaluation of the recursive algorithm shows that mean-squared
error (MSE) decays as 1/M^4 for an M-element frame, which is consistent with
previous results on optimal decay of MSE. Reconstruction using the canonical
dual frame is also studied, and several results relate properties of the
analysis frame to whether linear reconstruction techniques provide consistent
reconstructions.Comment: 29 pages, 5 figures; detailed added to proof of Theorem 4.3 and a few
minor correction
List-Decoding Gabidulin Codes via Interpolation and the Euclidean Algorithm
We show how Gabidulin codes can be list decoded by using a parametrization
approach. For this we consider a certain module in the ring of linearized
polynomials and find a minimal basis for this module using the Euclidean
algorithm with respect to composition of polynomials. For a given received
word, our decoding algorithm computes a list of all codewords that are closest
to the received word with respect to the rank metric.Comment: Submitted to ISITA 2014, IEICE copyright upon acceptanc
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