Binary extended perfect codes of length 16 and rank 14

All extended binary perfect (16, 4, 2¹¹) codes of rank 14 over the field F ₂ are classified. It is proved that among all nonequivalent extended binary perfect (16, 4, 2¹¹) codes there are exactly 1719 nonequivalent codes of rank 14 over F ₂. Among these codes there are 844 codes classified by Phelps (Solov’eva-Phelps codes) and 875 other codes obtained by the construction of Etzion-Vardy and by a new general doubling construction, presented in the paper. Thus, the only open question in the classification of extended binary perfect (16,4,2¹¹) codes is that on such codes of rank 15 over F ₂.     

Product perfect codes and steganography

A new coding technique to be used in steganography is evaluated. The performance of this new technique is computed and comparisons with the well-known theoretical upper bound, Hamming upper bound and basic LSB are established.     

Binary generalized (L,G) codes that are perfect in a weighted hamming metric

We propose a class of binary generalized (L,G) codes that are perfect in a weighted Hamming metric.     

Quotients of Gaussian graphs and their application to perfect codes

A graph-based model of perfect two-dimensional codes is presented in this work. This model facilitates the study of the metric properties of the codes. Signal spaces are modeled by means of Cayley graphs defined over the Gaussian integers and denoted as Gaussian graphs. Codewords of perfect codes will be represented by vertices of a quotient graph of the Gaussian graph in which the signal space has been defined. It will be shown that any quotient graph of a Gaussian graph is indeed a Gaussian graph. This makes it possible to apply previously known properties of Gaussian graphs to the analysis of perfect codes. To illustrate the modeling power of this graph-based tool, perfect Lee codes will be analyzed in terms of Gaussian graphs and their quotients.     

Local and interweight spectra of completely regular codes and of perfect colorings

We introduce notions of local and interweight spectra of an arbitrary coloring of a Boolean cube, which generalize the notion of a weight spectrum. The main objects of our research are colorings that are called perfect. We establish an interrelation of local spectra of such a coloring in two orthogonal faces of a Boolean cube and study properties of the interweight spectrum. Based on this, we prove a new metric property of perfect colorings, namely, their strong distance invariance. As a consequence, we obtain an analogous property of an arbitrary completely regular code, which, together with his neighborhoods, forms a perfect coloring.     

Construction of perfect q-ary codes by switchings of simple components

We suggest a construction of perfect q-ary codes by sequential switchings of special-type components (called simple components) of the Hamming code. We prove that such components are minimal. The construction yields a lower bound on the number of different q-ary codes; this is presently the best known bound. We show that this bound cannot be substantially improved using switchings of components of this type.     

On the constructions of constant-composition codes from perfect nonlinear functions

A new construction of constant-composition codes based on all known perfect nonlinear functions from F _q m to itself is presented, which provides a kind of unified constructions of constant-composition codes based on all known perfect nonlinear functions from F _q m to itself. It is proved that the new constant-composition codes are optimal with respect to the Luo-Fu-Vinck-Chen bound, when m is an odd positive integer greater than 1. Finally, we point out that two constructions of constant-composition codes, proposed by Ding Cunsheng et al. in 2005, are equivalent to two special types of the new constant-composition codes. Supported in part by the National Natural Science Foundation of China (Grant Nos. 60573028, 60803156), the Open Research Fund of the National Mobile Communications Research Laboratory of Southeast University (Grant No. W200805) and in part by Singapore Ministry of Education Academic Research Fund (Grant No. T206B2204)     

Volumes 15 and 16

Authors Index

Volumes 15 and 16     

Construction of punctured and extended quantum codes over GF(2)

On the basis of elementary transformation,we propose a new method for constructing a class of pure quantum codes [[n-i,2k-n+i,d-i]]2 and [[n+1,2k-n-1,d+1]]2 from a class of classical linear codes [n,k,d]2 that contain their dual codes.The construction process was based on the elementary algebra;the error-correcting performance of the quantum codes was analyzed according to the relationship between the parity-check matrix and the minimum distance of the classical linear codes;the encoding and decoding networks were constructed based on the stabilizer.The proposed method is simple,straightforward and easy to implement using a computer and other hardware system.The theoretical results showed that the method is practical for the construction of a class of quantum codes.     

Performance of punctured convolutional codes in asynchronous CDMA communications under perfect phase-tracking conditions

In this paper, the performance of punctured convolutional codes of short constraint lengths is discussed. The punctured codes are used to provide error protection to a particular user in an asynchronous code division multiple access (A-CDMA) system. Perfect channel estimation is assumed at the receiver. A slow fading Rician or Rayleigh channel is assumed. Maximum likelihood decoding through a Viterbi algorithm is used to decode the received symbols. Soft decision decoding for perfect phase tracking of the received signal is considered. Analytical bounds, which are useful in predicting the performance of the A-CDMA system are derived and plotted for the cases of infinite and finite channel memory. The upper bounds with Viterbi decoding are derived and plotted for the various punctured codes considered. The simulated results are found to agree very well with their upper bounds and predicted results.     

Fast and low-complexity encoding of Raptor codes based on operation lists for known source block lengths

Raptor codes can provide good error correction capability and efficient encoding and decoding rate. Its fountain code property is effective in avoiding packet retransmission for both unicast and multicast service delivery. Hence, there are many mobile, vehicular and broadband applications adopting it such as mobile multimedia broadcasting, multimedia communications for high-speed rails and broadband IPTV systems. In this paper, we present an efficient systematic Raptor codes encoder based on operation lists for known source block lengths. For a Raptor codes application that can frequently use one or several fixed source block lengths (i.e., the number of source symbols in a source block), we could produce the corresponding operation lists in advance and use them to generate the encoding symbols more efficiently. We first introduce the basic architecture of the proposed Raptor encoder, and then describe the details about how to generate operation lists and Raptor intermediate symbols. The simulation results show that our encoder is at least two times faster than the conventional Raptor codes encoder which is adopted by 3GPP and DVB-H. Besides, the reduction on CPU utilization for a real Raptor-based streaming server is described, which is from 11.59 % to 53.41 %, depending on the employed source block length and symbol size.     

Cardinality spectra of components of correlation immune functions,bent functions,perfect colorings,and codes

We study cardinalities of components of perfect codes and colorings, correlation immune functions, and bent function (sets of ones of these functions). Based on results of Kasami and Tokura, we show that for any of these combinatorial objects the component cardinality in the interval from 2 ^k to 2 ^k+1 can only take values of the form 2 ^k+1 ? 2 ^p , where p ∈ {0, ..., k} and 2 ^k is the minimum component cardinality for a combinatorial object with the same parameters. For bent functions, we prove existence of components of any cardinality in this spectrum. For perfect colorings with certain parameters and for correlation immune functions, we find components of some of the above-given cardinalities.     

多进制低密度奇偶校验码的扩展最小和译码算法研究

综述了多进制LDPC码的几种常用译码算法,重点讲解分析了其中的扩展最小和算法,并采用对比的方法证明其优越性。