Is the class of cyclic codes asymptotically good?

There is the long-standing question whether the class of cyclic codes is asymptotically good. By an old result of Lin and Weldon, long Bose-Chaudhuri-Hocquenhem (BCH) codes are asymptotically bad. Berman proved that cyclic codes are asymptotically bad if only finitely many primes are involved in the lengths of the codes. We investigate further classes of cyclic codes which also turn out to be asymptotically bad. Based on reduction arguments we give some evidence that there are asymptotically good sequences of binary cyclic codes in which all lengths are prime numbers provided there is any asymptotically good sequence of binary cyclic codes.     

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.      

Asymptotically Gaussian weight distribution and performance of multicomponent turbo block codes and product codes

It was suggested by Battail that a good long linear code should have a weight distribution close to that of random coding, rather than a large minimum distance, and a turbo code should be also designed using a random-like criterion. In this paper, we first show that the weight distribution of a high-rate linear block code is approximately Gaussian if the code rate is close enough to one, and then proceed to construct a low-rate linear block code with approximately Gaussian weight distribution by using the turbo-coding technique. We give a sufficient condition under which the weight distribution of multicomponent turbo block (MCTB) codes (multicomponent product (MCP) codes, respectively) can approach asymptotically that of random codes, and further develop two classes of MCTB codes (MCP codes) satisfying this condition. Simulation results show that MCTB codes (MCP codes) having asymptotically Gaussian weight distribution can asymptotically approach Shannon's capacity limit. MCTB codes based on single parity-check (SPC) codes have a far poorer minimum distance than MCP codes based on SPC codes, but we show by simulation that when the bit-error rate is in the important range of 10/sup -1/-10/sup -5/, these codes can still offer similar performance for the additive white Gaussian noise channel, as long as the code length of the SPC codes is not very short. These facts confirm in a more precise way Battail's inference about the "nonimportance" of the minimum distance for a long code.     

LDPC码在对流层散射通信中的性能分析

阐述了对流层散射信道恶劣的传输环境,表明它的衰落特性。介绍了LDPC码的基本原理,分析了迭代译码的更新过程以及LDPC码的内交织性。为了在对流层散射通信中有效应用LDPC码,研究了信道特性与编码性能之间的关系,对LDPC码在散射通信中的性能进行了仿真,结果表明,在对流层散射通信中,信道的衰落相关性对编码性能有非常重要的影响,给出了相应的参考曲线,为LDPC码的应用提供先验信息。     

Application of efficient Chase algorithm in decoding of generalized low-density parity-check codes

We consider the iterative decoding of generalized low-density (GLD) parity-check codes where, rather than employ an optimal subcode decoder, a Chase (1972) algorithm decoder more commonly associated with "turbo product codes" is used. GLD codes are low-density graph codes in which the constraint nodes are other than single parity-checks. For extended Hamming-based GLD codes, we use bit error rates derived by simulation to demonstrate this new strategy to be successful at higher code rates. For long block lengths, good performance close to capacity is possible with decoding costs reduced further since the Chase decoder employed is an efficient implementation.     

The serial concatenation of rate-1 codes through uniform random interleavers

Until the analysis of repeat accumulate codes by Divsalar et al. (1998), few people would have guessed that simple rate-1 codes could play a crucial role in the construction of "good" binary codes. We construct "good" binary linear block codes at any rate r<1 by serially concatenating an arbitrary outer code of rate r with a large number of rate-1 inner codes through uniform random interleavers. We derive the average output weight enumerator (WE) for this ensemble in the limit as the number of inner codes goes to infinity. Using a probabilistic upper bound on the minimum distance, we prove that long codes from this ensemble will achieve the Gilbert-Varshamov (1952) bound with high probability. Numerical evaluation of the minimum distance shows that the asymptotic bound can be achieved with a small number of inner codes. In essence, this construction produces codes with good distance properties which are also compatible with iterative "turbo" style decoding. For selected codes, we also present bounds on the probability of maximum-likelihood decoding (MLD) error and simulation results for the probability of iterative decoding error.     

Limits to List Decoding Reed&#8211;Solomon Codes

In this paper, we prove the following two results that expose some combinatorial limitations to list decoding Reed-Solomon codes. 1) Given n distinct elements alpha₁,...,alpha_n from a field F, and n subsets S₁,...,S_n of F, each of size at most l, the list decoding algorithm of Guruswami and Sudan can in polynomial time output all polynomials p of degree at most k that satisfy p(alpha_i)isinS_i for every i, as long as ldelta for small enough delta, we exhibit an explicit received word with a superpolynomial number of Reed-Solomon codewords that agree with it on (2-epsi)k locations, for any desired epsi>0 (agreement of k is trivial to achieve). Such a bound was known earlier only for a nonexplicit center. Finding explicit bad list decoding configurations is of significant interest-for example, the best known rate versus distance tradeoff, due to Xing, is based on a bad list decoding configuration for algebraic-geometric codes, which is unfortunately not explicitly known     

Computer codes for EMP interaction and coupling

The properties of a number of digital computer codes appropriate for various classes of external coupling problems are described. Limitations and approximations of numerical methods in general are reviewed. Thin-wire codes are tabulated to indicate their basic properties, special features, and structures analyzed, including grid models of surfaces and wire stick models of aircraft. Surface codes are tabulated for bodies of revolution, arbitrary surfaces and hybrid surface-wire or surface-aperture configurations in or below the "resonance" regime, and GTD codes for higher frequencies. Various aperture codes for studying apertures in planes, in empty bodies, or with a wire pickup behind the aperture ate summarized and tabulated. Some codes for long cables above or below the earth are reviewed. Conclusions warn against imprudent use of computer codes in general.     

Construction of (2k,k, 2) Convolutional Codes with Double-Loop Cyclic Codes

In this paper, by taking multiple-time information in blocks into the coding of linear block codes, a new class of (2 \(k\) , \(k\) , 2) convolutional codes is constructed, by which a new way of constructing long codes with short ones is obtained. After that, the type of embedded codes is determined and the optimal values of the linear combination coefficients are derived by using a three-dimensional state transfer matrix to analyze and testify the constructing mechanism of the codes. Finally, the simulation experiment tests the error-correcting performance of the (2 \(k\) , \(k\) , 2) convolutional codes for different value of \(k\) , it is shown that the performance of the new convolutional codes compares favorably with that of traditional (2, 1, \(l\) ) convolutional codes.     

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     

CRT-Based High-Speed Parallel Architecture for Long BCH Encoding

Bose–Chaudhuri–Hocquenghen (BCH) error-correcting codes are now widely used in communication system and digital technology. The direct linear feedback shifted register (LFSR)-based encoding of a long BCH code suffers from the large fan-out effect of some xor gates. This makes the LFSR-based encoders of long BCH codes not keep up with the data transmission speed in some applications. The technique for eliminating the large fan-out effect by J-unfolding method and some algebraic manipulation has been proposed. In this brief, we present a Chinese remainder theorem (CRT)-based parallel architecture for long BCH encoding. Our novel technique can be used to eliminate the fan-out bottleneck. The only restriction on the speed of long BCH encoding of our CRT-based architecture is $log_{2}N$, where $N$ is the length of the BCH code.      

Unequal error protection for convolutional codes

In this correspondence, unequal error-correcting capabilities of convolutional codes are studied. For errors in the information symbols and code symbols, the free input- and output-distances, respectively, serve as "unequal" counterparts to the free distance. When communication takes place close to or above the channel capacity the error bursts tend to be long and the free distance is not any longer useful as the measure of the error correcting capability. Thus, the active burst distance for a given output and the active burst distance for a given input are introduced as "unequal" counterparts to the active burst distance and improved estimates of the unequal error-correcting capabilities of convolutional codes are obtained and illustrated by examples. Finally, it is shown how to obtain unequal error protection for both information and code symbols using woven convolutional codes.     

Unitary Nongroup STBC From Cyclic Algebras

Space-time block codes (STBCs) are designed for multiple-input-multiple-output (MIMO) channels. In order to avoid errors, single-input-single-output (SISO) fading channels require long coding blocks and interleavers that result in high delays. If one wishes to increase the data rate it is necessary to take advantage of space diversity. Early STBC, that where developed by Alamouti for known channels and by Tarokh for unknown channels, have been proven to increase the performance of channels characterized by Rayleigh fading. Codes that are based on division algebras have by definition nonzero diversity and therefore are suitable for STBC in order to achieve high rates at low symbol-to-noise ratio (SNR). This work presents new high-diversity group-based STBCs with improved performance both in known and unknown channels. We describe two new sets of codes for multiple antenna communication. The first set is a set of "superquaternions" and improves considerably on the Alamouti codes. It is based on the mathematical fact that "normalized" integral quaternions are very well distributed over the unit sphere in four-dimensional (4-D) Euclidean space. The second set of codes gives arrays of 3times3 unitary matrices with full diversity. Here the idea is to use cosets of finite subgroups of division algebras that are nine-dimensional (9-D) over their center, which is a finite cyclotomic extension of the field of rational numbers. It is shown that these codes outperform Alamouti and G _mr     

On the construction of group block codes

We consider the construction of group block codes, i.e., subgroups of Gⁿ, the n-fold direct product of a group G. Two concepts are introduced that make this construction similar to that of codes over gf(2). The first concept is that of an indecomposable code. The second is that of a parity-check matrix. As a result, group block codes over a decomposable Abelian group of exponent d_m can be seen as block codes over the ring of residues modulo d_m, and their minimum Hamming distance can be easily determined. We also prove that, under certain technical conditions, (n, k) systematic group block codes over non-Abelian groups are asymptotically bad, in the sense that their minimum Hamming distance cannot exceed [n/k].     

Short error-correcting codes

Theoretical bounds on the error-correcting capabilities of binary codes can be found for very long codes in terms of k/n and t/n. It is pointed out that these bounds are only valid for very long codes, and that the relationship between k/n and t/n is not a sufficient criterion for the practical value of short codes.     

Repeated use of codes for error detection five times is bad

The undetected error probabilities of codes when used repeatedly to transmit sequences of messages are studied. It is shown that for any code whose rate is greater than zero but less than one, the coding scheme obtained by the repeated use of the code is bad, if used five times or more, and improper if used four times or more. Furthermore, the repeated use of any linear code, over an alphabet greater than five, more than once, is bad and improper     

LP Decoding Corrects a Constant Fraction of Errors

We show that for low-density parity-check (LDPC) codes whose Tanner graphs have sufficient expansion, the linear programming (LP) decoder of Feldman, Karger, and Wainwright can correct a constant fraction of errors. A random graph will have sufficient expansion with high probability, and recent work shows that such graphs can be constructed efficiently. A key element of our method is the use of a dual witness: a zero-valued dual solution to the decoding linear program whose existence proves decoding success. We show that as long as no more than a certain constant fraction of the bits are flipped by the channel, we can find a dual witness. This new method can be used for proving bounds on the performance of any LP decoder, even in a probabilistic setting. Our result implies that the word error rate of the LP decoder decreases exponentially in the code length under the binary-symmetric channel (BSC). This is the first such error bound for LDPC codes using an analysis based on "pseudocodewords." Recent work by Koetter and Vontobel shows that LP decoding and min-sum decoding of LDPC codes are closely related by the "graph cover" structure of their pseudocodewords; in their terminology, our result implies that that there exist families of LDPC codes where the minimum BSC pseudoweight grows linearly in the block length     

SPC乘积码用于纠删译码的性能分析

SPC乘积码是一种既可以纠错,又可以恢复删除的高效码。该码的最小码距是4,能够恢复所有单阶,两阶和三阶的删除模式,实际上这种码能够恢复更高阶的删除模式。传统上是通过最小码间距来评价SPC乘积码译码性能,若从其空间结构入手,能够推导出SPC乘积码迭代译码后删除率的新上界。仿真显示这个上界更紧。     

Suboptimum decoding of decomposable block codes

To decode a long block code with a large minimum distance by maximum likelihood decoding is practically impossible because the decoding complexity is simply enormous. However, if a code can be decomposed into constituent codes with smaller dimensions and simpler structure, it is possible to devise a practical and yet efficient scheme to decode the code. This paper investigates a class of decomposable codes, their distance and structural properties. It is shown that this class includes several classes of well-known and efficient codes as subclasses. Several methods for constructing decomposable codes or decomposing codes are presented. A two-stage (soft-decision or hard-decision) decoding scheme for decomposable codes, their translates or unions of translates is devised, and its error performance is analyzed for an AWGN channel. The two-stage soft-decision decoding is suboptimum. Error performances of some specific decomposable codes based on the proposed two-stage soft-decision decoding are evaluated. It is shown that the proposed two-stage suboptimum decoding scheme provides an excellent trade-off between the error performance and decoding complexity for codes of moderate and long block length     

On existence of good self-dual quasicyclic codes

For a long time, asymptotically good self-dual codes have been known to exist. Asymptotically good 2-quasicyclic codes of rate 1/2 have also been known to exist for a long time. Recently, it was proved that there are binary self-dual n/3-quasicyclic codes of length n asymptotically meeting the Gilbert-Varshamov bound. Unlike 2-quasicyclic codes, which are defined to have a cyclic group of order n/2 as a subgroup of their permutation group, the n/3-quasicyclic c codes are defined with a permutation group of fixed order of 3. So, from the decoding point of view, 2-quasicyclic c codes are preferable to n/3-quasicyclic c codes. In this correspondence, with the assumption that there are infinite primes p with respect to (w r t.) which 2 is primitive, we prove that there exist classes of self-dual 2p-quasicyclic c codes and Type II 8p-quasicyclic c codes of length respectively 2p/sup 2/ and 8p/sup 2/ which asymptotically meet the Gilbert-Varshamov bound. When compared with the order of the defining permutation groups, these classes of codes lie between the 2-quasicyclic c codes and the n/3-quasicyclic c codes of length n, considered in previous works.