Binary linear quasi-perfect codes are normal

Whether quasi-perfect codes are normal is addressed. Let C be a code of length n, dimension k, covering radius R, and minimal distance d. It is proved that C is normal if d⩾2R-1. Hence all quasi-perfect codes are normal. Consequently, any [n,k ]R binary linear code with minimal distance d⩾2R-1 is normal     

Binary image synthesis using mixed linear integer programming

Images that are to be transmitted through a distorting imaging system may be deliberately altered to compensate for that distortion. The authors consider an incoherent diffraction-limited imaging system followed by an ideal high-contrast detector that prints binary images, and seek a binary input image (a mask) that generates a desired prescribed binary output image. They have developed a method for determining the optimal binary mask by formulating the problem as a mixed linear integer program (MLIP) and using the branch and bound method to solve it.     

Binary lattice vector quantization with linear block codes andaffine index assignments

We determine analytic expressions for the performance of some low-complexity combined source-channel coding systems. The main tool used is the Hadamard transform. In particular, we obtain formulas for the average distortion of binary lattice vector quantization with affine index assignments, linear block channel coding, and a binary-symmetric channel. The distortion formulas are specialized to nonredundant channel codes for a binary-symmetric channel, and then extended to affine index assignments on a binary-asymmetric channel. Various structured index assignments are compared. Our analytic formulas provide a computationally efficient method for determining the performance of various coding schemes. One interesting result shown is that for a uniform source and uniform quantizer, the natural binary code is never optimal for a nonsymmetric channel, even though it is known to be optimal for a symmetric channel     

Binary uniform codes

An algorithm to construct all linear binary uniform codes is given. The algorithm makes use of block codes due to Macdonald.     

一种基于改进线性规划的LDPC码混合译码算法

与基于消息迭代的置信传播译码相比,线性规划(linear programming,LP)译码分析有限长LDPC码性能更为有效。然而,传统LP译码算法运算量非常大,不利于系统实现。本文结合LDPC码校验矩阵的特点,去掉传统LP译码中不必要的约束,得到一种低复杂度LP内点译码算法。为了降低译码延时,将LP内点译码算法与置信传播译码算法结合,提出LDPC码混合译码算法。仿真结果表明,混合译码算法的误码性能优于传统LP译码和BP译码算法,而译码延时低于传统LP译码。     

Binary convolutional codes with application to magnetic recording

Calderbank, Heegard, and Ozarow [1] have suggested a method of designing codes for channels with intersymbol interference, such as the magnetic recording channel. These codes are designed to exploit intersymbol interference. The standard method is to minimize intersymbol interference by constraining the input to the channel using run-length limited sequences. Calderbank, Heegard, and Ozarow considered an idealized model of an intersymbol interference channel that leads to the problem of designing codes for a partial response channel with transfer function(1 - D^{N}) /2, where the channel inputs are constrained to bepm 1. This problem is considered here. Channel inputs are generated using a nontrivial coset of a binary convolutional code. The coset is chosen to limit the zero-run length of the output of the channel and so maintain clock synchronization. The minimum squared Euclidean distance between outputs corresponding to distinct inputs is bounded below by the free distance of a second convolutional code which we call the magnitude code. An interesting feature of the analysis is that magnitude codes that are catastrophic may perform better than those that are noncatastrophic.     

Binary multilevel coset codes based on Reed-Muller codes

We study the construction and decoding of binary multilevel coset codes. This construction, originally introduced by Blokh and Zyablov in 1974 and by Zinov'ev in 1976, shows remarkable analogies with most recent schemes of coded modulations. Basic elements of the construction are an inner code, head of a partition chain having suitable distance properties, and a set of outer codes, generally nonbinary. For each partition level there is an outer code whose alphabet has the same order of the partition: in this way it is possible to associate every partition subset to a code symbol. It is well known that these codes can be efficiently decoded by the so called “multistage decoding.” We show that good codes (in terms of performance/complexity) can be constructed using Reed-Muller (RM) codes as inner codes. To this aim RM codes are revisited in the framework of the above construction and decoding techniques. In particular we describe a family of decoders for RM codes which include Forney's (1988) and Hemmati's (1989) decoders as special cases. Finally, we present some examples of efficient binary codes based on RM codes, and assess their performance via computer simulation     

二元Duadic码及其等价类

丁存生等在文献[1]中构造了所有码长为素数p的二元Dudaic码,我们进一步在文献[2]中构造了所有码长为p^m的二元Duadic码。然而如何构造和计数任意码长为n=p1^m1p2^m2…pr^mr的二元Duadic码,至今仍是一个公开问题。我们在本文中完全解决了这个问题,并且还给出了不等价的Duadic码的计数。     

线性等距码与极大投射码

本文证明任意有限域上的一个线性等距码等价于一个极大投射码的重复码,从而给出了一般q元线性等距码的全部结构。     

Mixed-Q linear space-time codes

A new modulation method for linear space-time codes is proposed based on using constellations of different sizes for different symbols. It is shown that the proposed method significantly reduces the complexity of the sphere decoding algorithm. The complexity reduction is more pronounced in high-rate codes, where each code matrix carries a large number of symbols. We also show that the choice of constellation size provides a tradeoff between performance and complexity. Using this, some guidelines for choosing constellation size are presented. As one introduces more constellation disparity in the code, the complexity is further reduced, while the performance loss grows. Typically, a complexity reduction of one to two orders of magnitude can be achieved at the expense of about 3 dB coding gain. We suggest a simple modification in our design to reduce this loss to about 2 dB.     

Optimal linear identifying codes

Identifying codes can be used to locate malfunctioning processors. We say that a code C of length n is a linear (1,/spl les/l)-identifying code if it is a subspace of F/sub 2//sup n/ and for all X,Y/spl sube/F/sub 2//sup n/ such that |X|, |Y|/spl les/l and X/spl ne/Y, we have /spl cup//sub x/spl isin/X/(B(x)/spl cap/C)/spl ne//spl cup/y/spl isin/Y(B(y)/spl cap/C). Strongly (1,/spl les/l)-identifying codes are a variant of identifying codes. We determine the cardinalities of optimal linear (1,/spl les/l)-identifying and strongly (1,/spl les/l)-identifying codes in Hamming spaces of any dimension for locating any at most l malfunctioning processors.     

On applying molecular computation to binary linear codes

Adleman's (1994) successful solution of a seven-vertex instance of the NP-complete Hamiltonian directed path problem by a DNA algorithm initiated the field of biomolecular computing. In this correspondence, we describe DNA algorithms based on the sticker model to perform encoding, minimum-distance computation, and maximum-likelihood (ML) decoding of binary linear codes. We also discuss feasibility and limitations of the sticker algorithms     

Binary multilevel coset codes with interlevel coding

An improved multilevel coset coding scheme is proposed. By providing interlevel coding between adjacent blocks, the proposed scheme has a multilevel and single-stage coding structure. The coding rate of the proposed scheme can be increased without decreasing the minimum Hamming distance as compared to the traditional scheme.     

Stacking techniques for linear codes

A procedure for subspace stacking is proposed, and it is proved that the technique is equally successful for both linear codes and anticodes. It is demonstrated that families of errorcorrecting codes may be constructed using the proposed technique for stacking linear codes. Examples of such codes are given with rates better than the best known codes of identical Hamming distance and the same number of information digits.     

Decoding by linear programming

This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f/spl isin/R/sup n/ from corrupted measurements y=Af+e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the /spl lscr//sub 1/-minimization problem (/spl par/x/spl par//sub /spl lscr/1/:=/spl Sigma//sub i/|x/sub i/|) min(g/spl isin/R/sup n/) /spl par/y - Ag/spl par//sub /spl lscr/1/ provided that the support of the vector of errors is not too large, /spl par/e/spl par//sub /spl lscr/0/:=|{i:e/sub i/ /spl ne/ 0}|/spl les//spl rho//spl middot/m for some /spl rho/>0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work. Finally, underlying the success of /spl lscr//sub 1/ is a crucial property we call the uniform uncertainty principle that we shall describe in detail.     

The category of linear codes

Slepian (1960) introduced a structure theory for linear, binary codes and proved that every such code was uniquely the sum of indecomposable codes. He had hoped to produce a canonical form for the generator matrix of an indecomposable code so that he might read off the properties of the code from such a matrix, but such a program proved impossible. We here work over an arbitrary field and define a restricted class of indecomposable codes-which we call critical. For these codes there is a quasicanonical form for the generator matrix. Every indecomposable code has a generator matrix that is obtained from the generator matrix of a critical, indecomposable code by augmentation. As an application of the this quasicanonical form we illuminate the perfect linear codes, giving, for example, a “canonical” form for the generator matrix of the ternary Golay (1949) code     

用误码率测试仪测试驱动FEC代码

纠错技术是设计现代数字通信系统的重要环节.也许,选择一种会大大降低设计速度并增加设计复杂度的纠错码并非难事,但结果往往却会令人失望.而选择合适的又是很高效的纠错码也很容易一一但要使用合适的测试仪器来帮忙.     

Linear programming bounds for doubly-even self-dual codes

Using a variant of the linear programming method we derive a new upper bound on the minimum distance d of doubly-even self-dual codes of length n. Asymptotically, for n growing, it gives d/n⩽0.166315···+o(1), thus improving on the Mallows-Odlyzko-Sloane bound of 1/6. To establish this, we prove that in any doubly even-self-dual code the distance distribution is asymptotically upper-bounded by the corresponding normalized binomial distribution in a certain interval     

Binary multilevel convolutional codes with unequal error protectioncapabilities

Binary multilevel convolutional codes (CCs) with unequal error protection (UEP) capabilities are studied. These codes belong to the class of generalized concatenated (GC) codes. Binary CCs are used as outer codes. Binary linear block codes of short length, and selected subcodes in their two-way subcode partition chain, are used as inner codes. Multistage decodings are presented that use Viterbi decoders operating on trellises with similar structure to that of the constituent binary CCs. Simulation results of example binary two-level CC's are also reported     

Binary codes for compound channels (Corresp.)

The error correcting capabilities are studied of some binary codes used on channels where both independent and burst errors may occur. The compound distance profile is introduced, and a lower bound on this profile is obtained for a class of modified maximum distance separable codes. Several constructions are presented as an illustration.