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A simple tight bound on error probability of block codes with application to turbo codes
 TMO Progr. Rep
, 1999
"... A simple bound on the probability of decoding error for block codes is derived in closed form. This bound is based on the bounding techniques developed by Gallager. We obtained an upper bound both on the worderror probability and the biterror probability of block codes. The bound is simple, since ..."
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Cited by 61 (3 self)
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A simple bound on the probability of decoding error for block codes is derived in closed form. This bound is based on the bounding techniques developed by Gallager. We obtained an upper bound both on the worderror probability and the biterror probability of block codes. The bound is simple, since it does not require any integration or optimization in its final version. The bound is tight since it works for signaltonoise ratios (SNRs) very close to the Shannon capacity limit. The bound uses only the weight distribution of the code. The bound for nonrandom codes is tighter than the original Gallager bound and its new versions derived by Sason and Shamai and by Viterbi and Viterbi. It also is tighter than the recent simpler bound by Viterbi and Viterbi and simpler than the bound by Duman and Salehi, which requires twoparameter optimization. For long blocks, it competes well with more complex bounds that involve integration and parameter optimization, such as the tangential sphere bound by Poltyrev, elaborated by Sason and Shamai, and investigated by Viterbi and Viterbi, and the geometry bound by Dolinar, Ekroot, and Pollara. We also obtained a closedform expression for the minimum SNR threshold that can serve as a tight upper bound on maximumlikelihood capacity of nonrandom codes. We also have shown that this minimum SNR threshold of our bound is the same as for the tangential sphere bound of Poltyrev. We applied this simple bound to turbolike codes. I.
Lower bounds on the error probability of block codes based on improvements on de Caen’s inequality
 IEEE TRANS. INFORM. THEORY
, 2004
"... New lower bounds on the error probability of block codes with maximumlikelihood decoding are proposed. The bounds are obtained by applying a new lower bound on the probability of a union of events, derived by improving on de Caen’s lower bound. The new bound includes an arbitrary function to be op ..."
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Cited by 14 (0 self)
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New lower bounds on the error probability of block codes with maximumlikelihood decoding are proposed. The bounds are obtained by applying a new lower bound on the probability of a union of events, derived by improving on de Caen’s lower bound. The new bound includes an arbitrary function to be optimized in order to achieve the tightest results. Since the optimal choice of this function is known, but leads to a trivial and useless identity, we find several useful approximations for it, each resulting in a new lower bound. For the additive white Gaussian noise (AWGN) channel and the binarysymmetric channel (BSC), the optimal choice of the optimization function is stated and several approximations are proposed. When the bounds are further specialized to linear codes, the only knowledge on the code used is its weight enumeration. The results are shown to be tighter than the latest bounds in the current literature, such as those by Seguin and by Keren and Litsyn. Moreover, for the BSC, the new bounds widen the range of rates for which the union bound analysis applies, thus improving on the bound to the error exponent compared with the de Caenbased bounds.
Tightened upper bounds on the ML decoding error probability of binary linear block codes
 IEEE TRANS. ON INFORMATION THEORY
, 2006
"... The performance of maximumlikelihood (ML) decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability. The upper bounds on the block and bit error probabilities are valid for any memoryless, binaryinput and outputsymmetric communi ..."
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Cited by 13 (5 self)
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The performance of maximumlikelihood (ML) decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability. The upper bounds on the block and bit error probabilities are valid for any memoryless, binaryinput and outputsymmetric communication channel, and their effectiveness is exemplified for various ensembles of turbolike codes over the AWGN channel. An expurgation of the distance spectrum of binary linear block codes further tightens the resulting upper bounds.
Coded modulation in the blockfading channel: Coding theorems and code construction
 IEEE Trans. Inform. Theory
, 2006
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The partition weight enumerator of MDS codes and its applications
 in IEEE International Symposium on Information Theory
, 2005
"... AbstractA closed form formula of the partition weight enumerator of maximum distance separable (MDS) codes is derived for an arbitrary number of partitions. Using this result, some properties of MDS codes are discussed. The results are extended for the average binary image of MDS codes in finite f ..."
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Cited by 6 (2 self)
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AbstractA closed form formula of the partition weight enumerator of maximum distance separable (MDS) codes is derived for an arbitrary number of partitions. Using this result, some properties of MDS codes are discussed. The results are extended for the average binary image of MDS codes in finite fields of characteristic two. As an application, we study the multiuser error probability of Reed Solomon codes.
Performance Analysis of Turbo Codes in QuasiStatic Fading Channels
"... The performance of turbo codes in quasistatic fading channels both with and without antenna diversity is investigated. In particular, simple analytic techniques that relate the frame error rate of a turbo code to both its average distance spectrum as well as the iterative decoder convergence charac ..."
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Cited by 6 (1 self)
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The performance of turbo codes in quasistatic fading channels both with and without antenna diversity is investigated. In particular, simple analytic techniques that relate the frame error rate of a turbo code to both its average distance spectrum as well as the iterative decoder convergence characteristics are developed. Both by analysis and simulation, the impact of the constituent recursive systematic convolutional (RSC) codes, the interleaver size and the number of decoding iterations on the performance of turbo codes are also investigated. In particular, it is shown that in systems with limited antenna diversity different constituent RSC codes or interleaver sizes do not affect the performance of turbo codes. In contrast, in systems with significant antenna diversity, particular constituent RSC codes and interleaver sizes have the potential to significantly enhance the performance of turbo codes. These results are attributed to the fact that in single transmit–single receive antenna systems, the performance primarily depends on the decoder convergence characteristics for Eb/N0 values of practical interest. However, in multiple transmit–multiple receive antenna systems, the performance depends on the code characteristics.
The performance of serial turbo codes does not concentrate
 in "IEEE Transactions on Information Theory", 2011, accepted, http://hal.inria.fr/hal00641106/en. 28 Activity Report INRIA 2011
"... Abstract—Minimum distances and maximum likelihood error probabilities of serial turbo codes with uniform interleaver are analyzed. It is shown that, for a fraction of interleavers approaching one as the blocklength grows large, the minimum distance of serial turbo codes grows as a positive power ..."
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Cited by 2 (0 self)
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Abstract—Minimum distances and maximum likelihood error probabilities of serial turbo codes with uniform interleaver are analyzed. It is shown that, for a fraction of interleavers approaching one as the blocklength grows large, the minimum distance of serial turbo codes grows as a positive power of their blocklength, while their error probability decreases exponentially fast in some positive power of their blocklength, on sufficiently good memoryless channels. Such a typical code behavior contrasts the performance of the average serial turbo code, whose error probability is dominated by an asymptotically negligible fraction of poorly performing interleavers, and decays only as a negative power of the blocklength. The analysis proposed in this paper relies on precise bounds of the minimum distance of the typical serial turbo code, whose scaling law is shown to depend both on the free distance of its outer constituent encoder, which determines the exponent of its sublinear growth in the blocklength, and on the effective free distance of its inner constituent encoder. The latter is defined as the smallest weight of codewords obtained when the input word of the inner encoder has weight two, and appears as a linear scaling factor for the minimum distance of the typical serial turbo code. Hence, despite the lack of concentration of the maximum likelihood error probability around its expected value, the main design parameters suggested by the averagecode analysis turn out to characterize also the performance of the typical serial turbo code. By showing for the first time that the typical serial turbo code’s minimum distance scales linearly in the effective free distance of the inner constituent encoder, the presented results generalize, and improve upon, the probabilistic bounds of Kahale and Urbanke, as well as the deterministic upper bound of Bazzi, Mahdian, and Spielman, where only the dependence on the outer encoder’s free distance was proved. Index Terms—Error probability, minimum distance, serially concatenated codes, turbo codes, typical code analysis. I.
A new method for performance evaluation of bit decoding algorithms using statistics of the log likelihood ratio
, 2008
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A New Method for Performance Evaluation of TurboCodes Using . . .
 IEEE TRANSACTIONS ON COMMUNICATIONS
, 2004
"... In this paper, we present a new method for the performance evaluation of TurboCodes. The method ..."
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Cited by 2 (2 self)
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In this paper, we present a new method for the performance evaluation of TurboCodes. The method
Improved Tight Performance Bounds on Concatenated Codes
, 1999
"...  We derive new approximate upper bounds on the error probability of linear codes with maximum likelihood decoding by applying an appropriate transformation of decision variables. The new bounds are simple to calculate and based on the standard union bound but give tight results in some range below ..."
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Cited by 2 (0 self)
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 We derive new approximate upper bounds on the error probability of linear codes with maximum likelihood decoding by applying an appropriate transformation of decision variables. The new bounds are simple to calculate and based on the standard union bound but give tight results in some range below the channel cuto rate R 0 . In applying the bound to iterative decodable codes, such as turbo codes, we use the concept of uniform interleavers and determine the average distance spectrum. A comparison of our bounds to simulated error rates shows a signicantly improved tightness similar to the tangential sphere bound of Poltyrev, which is the best rigorous upper bound known up to now. Keywords Performance bounds, concatenated codes, turbo codes. I. Introduction Since the introduction of turbo codes in [3] there has been a growing interest and demand on calculating tight performance bounds of concatenated coding systems with interleaving in order to have a possibility to estimate the qu...