A logarithmic upper bound on the minimum distance of turbo codes

We derive new upper bounds on the minimum distance, which turbo codes can maximally attain with the optimum interleaver of a given length. The new bounds grow approximately logarithmically with the interleaver length, and they are tighter than all previously derived bounds for medium-length and long interleavers. An extensive discussion highlights the impacts of the new bounds in the context of interleaver design and provides some new design guidelines.     

Krawtchouk polynomials and universal bounds for codes and designsin Hamming spaces

Universal bounds for the cardinality of codes in the Hamming space F_rⁿ with a given minimum distance d and/or dual distance d' are stated. A self-contained proof of optimality of these bounds in the framework of the linear programming method is given. The necessary and sufficient conditions for attainability of the bounds are found. The parameters of codes satisfying these conditions are presented in a table. A new upper bound for the minimum distance of self-dual codes and a new lower bound for the crosscorrelation of half-linear codes are obtained     

Improved Performance Upper Bounds for Terminated Convolutional Codes

In this letter, we propose tight performance upper bounds for convolutional codes terminated with an input sequence of finite length. To obtain the upper bounds, a weight enumerator is defined to represent the relation between the Hamming distance of the coded output and the Hamming distance of the input bits of the code. The upper bounds on frame error rate (FER) and average bit error rate (BER) are obtained from the weight enumerator. A simple method is presented to compute the weight enumerator of a terminated convolutional code based on a modified trellis diagram.     

Improving the Upper Bounds on the Covering Radii of Binary Reed–Muller Codes

By deriving bounds on character sums of Boolean functions and by using the characterizations, due to Kasami , of those elements of the Reed-Muller codes whose Hamming weights are smaller than twice and a half the minimum distance, we derive an improved upper bound on the covering radius of the Reed-Muller code of order 2, and we deduce improved upper bounds on the covering radii of the Reed-Muller codes of higher orders     

Composition bounds for channel block codes

The performance of Channel block codes for a general channel is studied by examining the relationship between the rate of a code, the joint composition of pairs of codewords, and the probability of decoding error. At fixed rate, lower bounds and upper bounds, both on minimum Bhattacharyya distance between codewords and on minimum equivocation distance between codewords, are derived. These bounds resemble, respectively, the Gilbert and the Elias bounds on the minimum Hamming distance between codewords. For a certain large class of channels, a lower bound on probability of decoding error for low-rate channel codes is derived as a consequence of the upper bound on Bhattacharyya distance. This bound is always asymptotically tight at zero rate. Further, for some channels, it is asymptotically tighter than the straight line bound at low rates. Also studied is the relationship between the bounds on codeword composition for arbitrary alphabets and the expurgated bound for arbitrary channels having zero error capacity equal to zero. In particular, it is shown that the expurgated reliability-rate function for blocks of letters is achieved by a product distribution whenever it is achieved by a block probability distribution with strictly positive components.     

Laminated Turbo Codes: A New Class of Block-Convolutional Codes

In this paper, a new class of codes is presented that features a block-convolutional structure-namely, laminated turbo codes. It allows combining the advantages of both a convolutional encoder memory and a block permutor, thus allowing a block-oriented decoding method. Structural properties of laminated turbo codes are analyzed and upper and lower bounds on free distance are obtained. It is then shown that the performance of laminated turbo codes compares favorably with that of turbo codes. Finally, we show that laminated turbo codes provide high rate flexibility without suffering any significant performance degradation.     

Upper bound on the minimum distance of turbo codes

An upper bound on the minimum distance of turbo codes is derived, which depends only on the interleaver length and the component scramblers employed. The derivation of this bound considers exclusively turbo encoder input words of weight 2. The bound does not only hold for a particular interleaver but for all possible interleavers including the best. It is shown that in contrast to general linear binary codes the minimum distance of turbo codes cannot grow stronger than the square root of the block length. This implies that turbo codes are asymptotically bad. A rigorous proof for the bound is provided, which is based on a geometric approach     

A new upper bound on the minimum distance of turbo codes

In this paper, a new upper bound on the minimum distance of turbo codes is derived. The new bound is obtained by construction of an undirected graph which reflects the characteristics of the constituent codes and the interleaver. The resulting expression shows that the minimum distance of a turbo code grows approximately with the base-3 logarithm of the information word length. The new bound is easy to compute, applies to rate k/sub 0//n/sub 0/ constituent encoders, and often improves over existing results.     

The Minimum Distance of Turbo-Like Codes

Worst-case upper bounds are derived on the minimum distance of parallel concatenated turbo codes, serially concatenated convolutional codes, repeat-accumulate codes, repeat-convolute codes, and generalizations of these codes obtained by allowing nonlinear and large-memory constituent codes. It is shown that parallel-concatenated turbo codes and repeat-convolute codes with sub-linear memory are asymptotically bad. It is also shown that depth-two serially concatenated codes with constant-memory outer codes and sublinear-memory inner codes are asymptotically bad. Most of these upper bounds hold even when the convolutional encoders are replaced by general finite-state automata encoders. In contrast, it is proven that depth-three serially concatenated codes obtained by concatenating a repetition code with two accumulator codes through random permutations can be asymptotically good.      

Performance Analysis and Code Design for Minimum Hamming Distance Fusion in Wireless Sensor Networks

Distributed classification fusion using error-correcting codes (DCFECC) has recently been proposed for wireless sensor networks operating in a harsh environment. It has been shown to have a considerably better capability against unexpected sensor faults than the optimal likelihood fusion. In this paper, we analyze the performance of a DCFECC code with minimum Hamming distance fusion. No assumption on identical distribution for local observations, as well as common marginal distribution for the additive noises of the wireless links, is made. In addition, sensors are allowed to employ their own local classification rules. Upper bounds on the probability of error that are valid for any finite number of sensors are derived based on large deviations technique. A necessary and sufficient condition under which the minimum Hamming distance fusion error vanishes as the number of sensors tends to infinity is also established. With the necessary and sufficient condition and the upper error bounds, the relation between the fault-tolerance capability of a DCFECC code and its pair-wise Hamming distances is characterized, and can be used together with any code search criterion in finding the code with the desired fault-tolerance capability. Based on the above results, we further propose a code search criterion of much less complexity than the minimum Hamming distance fusion error criterion adopted earlier by the authors. This makes the code construction with acceptable fault-tolerance capability for a network with over a hundred of sensors practical. Simulation results show that the code determined based on the new criterion of much less complexity performs almost identically to the best code that minimizes the minimum Hamming distance fusion error. Also simulated and discussed are the performance trends of the codes searched based on the new simpler criterion with respect to the network size and the number of hypotheses     

Tailbiting codes obtained via convolutional codes with large active distance-slopes

The slope of the active distances is an important parameter when investigating the error-correcting capability of convolutional codes and the distance behavior of concatenated convolutional codes. The slope of the active distances is equal to the minimum average weight cycle in the state-transition diagram of the encoder. A general upper bound on the slope depending on the free distance of the convolutional code and new upper bounds on the slope of special classes of binary convolutional codes are derived. Moreover, a search technique, resulting in new tables of rate R=1/2 and rate R=1/3 convolutional encoders with high memories and large active distance-slopes is presented. Furthermore, we show that convolutional codes with large slopes can be used to obtain new tailbiting block codes with large minimum distances. Tables of rate R=1/2 and rate R=1/3 tailbiting codes with larger minimum distances than the best previously known quasi-cyclic codes are given. Two new tailbiting codes also have larger minimum distances than the best previously known binary linear block codes with same size and length. One of them is also superior in terms of minimum distance to any previously known binary nonlinear block code with the same set of parameters.     

Upper bounds on the rate of LDPC codes as a function of minimum distance

New upper bounds on the rate of low-density parity-check (LDPC) codes as a function of the minimum distance of the code are derived. The bounds apply to regular LDPC codes, and sometimes also to right-regular LDPC codes. Their derivation is based on combinatorial arguments and linear programming. The new bounds improve upon the previous bounds due to Burshtein et al. It is proved that at least for high rates, regular LDPC codes with full-rank parity-check matrices have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound.     

关于BCH码的广义Hamming重量上,下限

一个线性码的第ｒ广义Ｈａｍｍｉｎｇ重量是它任意ｒ维子码的最小支集大小。本文给出了一般（本原、狭义）ＢＣＨ码的广义Ｈａｍｍｉｎｇ重量下限和一类ＢＣＨ码的广义Ｈａｍｍｉｎｇ重量上限     

On the minimum distance of ternary cyclic codes

There are many ways to find lower bounds for the minimum distance of a cyclic code, based on investigation of the defining set. Some new theorems are derived. These and earlier techniques are applied to find lower bounds for the minimum distance of ternary cyclic codes. Furthermore, the exact minimum distance of ternary cyclic codes of length less than 40 is computed numerically. A table is given containing all ternary cyclic codes of length less than 40 and having a minimum distance exceeding the BCH bound. It seems that almost all lower bounds are equal to the minimum distance. Especially shifting, which is also done by computer, seems to be very powerful. For length 40⩽n⩽50, only lower bounds are computed. In many cases (derived theoretically), however, these lower bounds are equal to the minimum distance     

On general Gilbert bounds

We define a distance measure for block codes used over memoryless channels and show that it is related to upper and lower bounds on the low-rate error probability in the same way as Hamming distance is for binary block codes used over the binary symmetric channel. We then prove general Gilbert bounds for block codes using this distance measure. Some new relationships between coding theory and rate-distortion theory are presented.     

Woven codes with outer warp: variations, design, and distanceproperties

We consider convolutional and block encoding schemes which are variations of woven codes with outer warp. We propose methods to evaluate the distance characteristics of the considered codes on the basis of the active distances of the component codes. With this analytical bounding technique, we derived lower bounds on the minimum (or free) distance of woven convolutional codes, woven block codes, serially concatenated codes, and woven turbo codes. Next, we show that the lower bound on the minimum distance can be improved if we use designed interleaving with unique permutation functions in each row of the warp of the woven encoder. Finally, with the help of simulations, we get upper bounds on the minimum distance for some particular codes and then investigate their performance in the Gaussian channel. Throughout this paper, we compare all considered encoding schemes by means of examples, which illustrate their distance properties     

Improved upper bounds on the ML decoding error probability ofparallel and serial concatenated turbo codes via their ensemble distancespectrum

The ensemble performance of parallel and serial concatenated turbo codes is considered, where the ensemble is generated by a uniform choice of the interleaver and of the component codes taken from the set of time-varying recursive systematic convolutional codes. Following the derivation of the input-output weight enumeration functions of the ensembles of random parallel and serial concatenated turbo codes, the tangential sphere upper bound is employed to provide improved upper bounds on the block and bit error probabilities of these ensembles of codes for the binary-input additive white Gaussian noise (AWGN) channel, based on coherent detection of equi-energy antipodal signals and maximum-likelihood decoding. The influence of the interleaver length and the memory length of the component codes is investigated. The improved bounding technique proposed here is compared to the conventional union bound and to a alternative bounding technique by Duman and Salehi (1998) which incorporates modified Gallager bounds. The advantage of the derived bounds is demonstrated for a variety of parallel and serial concatenated coding schemes with either fixed or random recursive systematic convolutional component codes, and it is especially pronounced in the region exceeding the cutoff rate, where the performance of turbo codes is most appealing. These upper bounds are also compared to simulation results of the iterative decoding algorithm     

Convergence analysis of turbo decoding of product codes

Geometric interpretation of turbo decoding has founded an analytical basis, and provided tools for the analysis of this algorithm. We focus on turbo decoding of product codes, and based on the geometric framework, we extend the analytical results and show how analysis tools can be practically adapted for this case. Specifically, we investigate the algorithm's stability and its convergence rate. We present new results concerning the structure and properties of stability matrices of the algorithm, and develop upper bounds on the algorithm's convergence rate. We prove that for any 2×2 (information bits) product codes, there is a unique and stable fixed point. For the general case, we present sufficient conditions for stability. The interpretation of these conditions provides an insight to the behavior of the decoding algorithm. Simulation results, which support and extend the theoretical analysis, are presented for Hamming [(7,4,3)]² and Golay [(24,12,8)]² product codes     

New code upper bounds from the Terwilliger algebra and semidefinite programming

We give a new upper bound on the maximum size A(n,d) of a binary code of word length n and minimum distance at least d. It is based on block-diagonalizing the Terwilliger algebra of the Hamming cube. The bound strengthens the Delsarte bound, and can be calculated with semidefinite programming in time bounded by a polynomial in n. We show that it improves a number of known upper bounds for concrete values of n and d. From this we also derive a new upper bound on the maximum size A(n,d,w) of a binary code of word length n, minimum distance at least d, and constant weight w, again strengthening the Delsarte bound and yielding several improved upper bounds for concrete values of n, d, and w     

Multilevel turbo coding with short interleavers

The impact of the interleaver, embedded in the encoder for a parallel concatenated code, called the turbo code, is studied. The known turbo codes consist of long random interleavers, whose purpose is to reduce the value of the error coefficients. It is shown that an increased minimum Hamming distance can be obtained by using a structured interleaver. For low bit-error rates (BERs), we show that the performance of turbo codes with a structured interleaver is better than that obtained with a random interleaver. Another important advantage of the structured interleaver is the short length required, which yields a short decoding delay and reduced decoding complexity (in terms of memory). We also consider the use of turbo codes as component codes in multilevel codes. Powerful coding structures that consist of two component codes are suggested. Computer simulations are performed in order to evaluate the reduction in coding gain due to suboptimal iterative decoding. From the results of these simulations we deduce that the degradation in the performance (due to suboptimal decoding) is very small