573 research outputs found
Complexity Analysis of Reed-Solomon Decoding over GF(2^m) Without Using Syndromes
For the majority of the applications of Reed-Solomon (RS) codes, hard
decision decoding is based on syndromes. Recently, there has been renewed
interest in decoding RS codes without using syndromes. In this paper, we
investigate the complexity of syndromeless decoding for RS codes, and compare
it to that of syndrome-based decoding. Aiming to provide guidelines to
practical applications, our complexity analysis differs in several aspects from
existing asymptotic complexity analysis, which is typically based on
multiplicative fast Fourier transform (FFT) techniques and is usually in big O
notation. First, we focus on RS codes over characteristic-2 fields, over which
some multiplicative FFT techniques are not applicable. Secondly, due to
moderate block lengths of RS codes in practice, our analysis is complete since
all terms in the complexities are accounted for. Finally, in addition to fast
implementation using additive FFT techniques, we also consider direct
implementation, which is still relevant for RS codes with moderate lengths.
Comparing the complexities of both syndromeless and syndrome-based decoding
algorithms based on direct and fast implementations, we show that syndromeless
decoding algorithms have higher complexities than syndrome-based ones for high
rate RS codes regardless of the implementation. Both errors-only and
errors-and-erasures decoding are considered in this paper. We also derive
tighter bounds on the complexities of fast polynomial multiplications based on
Cantor's approach and the fast extended Euclidean algorithm.Comment: 11 pages, submitted to EURASIP Journal on Wireless Communications and
Networkin
Quasi-optimal multiplication of linear differential operators
We show that linear differential operators with polynomial coefficients over
a field of characteristic zero can be multiplied in quasi-optimal time. This
answers an open question raised by van der Hoeven.Comment: To appear in the Proceedings of the 53rd Annual IEEE Symposium on
Foundations of Computer Science (FOCS'12
Fast In-place Algorithms for Polynomial Operations: Division, Evaluation, Interpolation
We consider space-saving versions of several important operations on
univariate polynomials, namely power series inversion and division, division
with remainder, multi-point evaluation, and interpolation. Now-classical
results show that such problems can be solved in (nearly) the same asymptotic
time as fast polynomial multiplication. However, these reductions, even when
applied to an in-place variant of fast polynomial multiplication, yield
algorithms which require at least a linear amount of extra space for
intermediate results. We demonstrate new in-place algorithms for the
aforementioned polynomial computations which require only constant extra space
and achieve the same asymptotic running time as their out-of-place
counterparts. We also provide a precise complexity analysis so that all
constants are made explicit, parameterized by the space usage of the underlying
multiplication algorithms
On the complexity of skew arithmetic
13 pagesIn this paper, we study the complexity of several basic operations on linear differential operators with polynomial coefficients. As in the case of ordinary polynomials, we show that these complexities can be expressed in terms of the cost of multiplication
Faster relaxed multiplication
In previous work, we have introduced several fast algorithms for relaxed power series multiplication (also known under the name on-line multiplication) up till a given order n. The fastest currently known algorithm works over an effective base field K with sufficiently many 2^p-th roots of unity and has algebraic time complexity O(n log n exp (2 sqrt (log 2 log log n))). In this note, we will generalize this algorithm to the cases when K is replaced by an effective ring of positive characteristic or by an effective ring of characteristic zero, which is also torsion-free as a Z-module and comes with an additional algorithm for partial division by integers. We will also present an asymptotically faster algorithm for relaxed multiplication of p-adic numbers
Elliptic periods for finite fields
We construct two new families of basis for finite field extensions. Basis in
the first family, the so-called elliptic basis, are not quite normal basis, but
they allow very fast Frobenius exponentiation while preserving sparse
multiplication formulas. Basis in the second family, the so-called normal
elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We
prove that all extensions admit models of this kind
Withdrawn paper: fast multiplication of integer matrices
THIS PAPER HAS BEEN WITHDRAWN. We briefly discuss the error which was made in the original version of the withdrawn paper. Original abstract: in this paper we will show that dense n×n matrices with integer coefficients of bit sizes ⩽b can be multiplied in quasi-optimal time. This shows that the exponent ω_ℤ for matrix multiplication over ℤ is equal to two. Moreover, there is hope that the exponent can be observed in practice for a sufficiently good implementation
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