Number of information symbols in polynomial codes

Polynomial codes and their dual codes as introduced by Kasami, Lin, and Peterson have considerable algebraic and geometric structure. It has been shown that these codes contain many well-known classes of cyclic codes as subclasses, such as BCH codes, projective geometry codes (PG codes), Euclidean geometry codes (EG codes), and generalized Reed-Muller codes (GRM codes). In this paper, combinatorial expressions for the number of information symbols and parity-check symbols in polynomial codes are derived. The results are applied to two important subclasses of codes, the PG codes and EG codes.     

一种三元线性互补对偶码与自正交码的构造方法

有限域上线性互补对偶(LCD)码有良好的相关特性和正交特性,并能够防御信道攻击。自正交码是编码理论中一类非常重要的码,可以用于构造量子纠错码。该文研究了有限域F3上的LCD码。通过选取4种合适的定义集,利用有限域F3上线性码是LCD码或自正交码的判定条件,构造了4类3元LCD码和一些自正交码,并研究了这4类线性码的对偶码,得到了一些3元最优线性码。     

Generalized tensor product codes

A class of binary error-correcting codes, called generalized tensor product codes, is presented with their decoding algorithm. These codes are constructed by combining a number of codes on various extension fields with shorter binary codes. A general algorithm is provided to do bounded distance decoding for these codes. Simply decodable codes such as Wolf's tensor product codes are shown to be special cases of this class of codes. Simply decodable and more efficient codes than Wolf's codes are also included as special cases.     

Application of Error-Correcting Codes in Computer Reliability Studies

The use of error-correcting codes as one of the important techniques to increase computer system reliability is introduced. The different codes used in the central processing unit (CPU) are described. Since the CPU usually contains the data path, logic, and arithmetic units, the codes used in this area are error-detecting codes, such as parity check codes and residue codes. The codes used or suggested for the memory system are discussed, emphasis being placed on parity check codes, two-dimensional codes, Hamming codes and other recently developed codes. The various codes used in the input/output system are presented. The input/output area of the computer system is relatively unreliable as compared with CPU or memory; therefore, error-correcting codes used in this area usually are much more powerful than single parity check codes. These include codes for the magnetic tape, disk, and drum units. The error coding techniques are compared with other techniques for increasing computer system reliability. The future trend of using error-correcting codes in a computer system is also discussed.     

一种四元厄米特LCD码与厄米特自正交码的构造方法

有限域上线性互补对偶（LCD）码具有良好的结构和性质,并在双用户加法器信道中得到了广泛的应用.自正交码是编码理论中一类重要的线性码,常被用于构造量子纠错码.本文根据有限域上线性码是厄米特LCD码或厄米特自正交码的判定条件,通过选取合适的定义集,构造出了四类四元厄米特LCD码和厄米特自正交码.同时,本文还研究了这四类线性码的厄米特对偶码,并得到了一些四元最优线性码.     

On majority-logic decoding for duals of primitive polynomial codes

The class of polynomial codes introduced by Kasami et al. has considerable inherent algebraic and geometric structure. It has been shown that this class of codes and their dual codes contain many important classes of cyclic codes as subclasses, such as BCH codes, Reed-Solomon codes, generalized Reed-Muller codes, projective geometry codes, and Euclidean geometry codes. The purpose of this paper is to investigate further properties of polynomial codes and their duals. First, majority-logic decoding for the duals of certain primitive polynomial codes is considered. Two methods of forming nonorthogonal parity-check sums are presented. Second, the maximality of Euclidean geometry codes is proved. The roots of the generator polynomial of an Euclidean geometry code are specified.     

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.      

A General Construction of Inter-Group Complementary Codes Based on Z-Complementary Codes and Perfect Periodic Cross-Correlation Codes

In this paper, a new construction of inter-group complementary (IGC) codes is proposed based on Z-complementary codes and perfect periodic cross-correlation (PPCC) codes. The correlation properties of the new generated IGC codes are as follows: (1) the autocorrelation sidelobe values of the codes are zeros in the zero correlation zone (ZCZ), (2) the cross-correlation values between different codes in the same group are zeros in the ZCZ, (3) the cross-correlation values between codes of different groups are zeros during the whole period. Comparing to the existed IGC codes, the ZCZ length of the generated IGC codes can be predefined flexibly; the number of groups and the number of codes in each group have more freedom according to the advantages of Z-complementary codes and PPCC codes.     

无速率编码及其应用综述

纠错编码技术通过引入冗余增加可靠性,是现代通信的关键技术之一。无速率编码是一类新兴纠错编码,其速率可以根据信道状态自适应改变,编译码算法较为简单,且性能优异,可以适用于不同的应用场景,因此受到了国内外学者和工业界的关注。介绍了4种经典或新兴的无速率编码方案,包括卢比变换(Luby Transform,LT)码、Raptor码、在线喷泉码(OFC)和BATS(Batched Sparse)码。介绍无速率编码的基本原理,通过其发展过程比较不同无速率编码的特点。阐述了这些无速率编码的编译码方法,并简要介绍其最新的研究进展。最后,介绍无速率编码在广播通信及不等差保护、无线传感器网络、车联网、存储以及分布式计算等新老场景中的应用。无速率编码是一种复杂度低、灵活度高的编码,随着新型无速率编码的发展,在未来的分布式系统等场景中将会有更广泛的应用。     

Polynomial codes

A class of cyclic codes is introduced by a polynomial approach that is an extension of the Mattson-Solomon method and of the Muller method. This class of codes contains several important classes of codes as subclasses, namely, BCH codes, Reed-Solomon codes, generalized primitive Reed-Muller codes, and finite geometry codes. Certain fundamental properties of this class of codes are derived. Some subclasses are shown to be majority-logic decodable.     

On the Performance of Short Forward Error-Correcting Codes

This letter investigates the performance of short forward error-correcting (FEC) codes. Reed-Solomon (RS) codes and concatenated zigzag codes are chosen as representatives of classical algebraic codes and modern simple iteratively decodable codes, respectively. Additionally, random binary linear codes are used as a baseline reference. Our main results (demonstrated by simulations and ensemble distance spectrum analysis) are as follows: 1) Short RS codes are as good as random binary linear codes; 2) Carefully designed short low-density parity-check (LDPC) codes are almost as good as random binary linear codes; 3) Low complexity belief propagation decoders incur considerable performance loss at short coding lengths. Thus, future work could focus on developing low-complexity (near) optimal decoders for RS codes and/or LDPC codes.     

一种基于之型分量码的系统GLDPC码

本文以规则低密度生成矩阵码为基础,构建了一种以之型码为分量码的系统广义低密度奇偶校验(Generalized Low-Density Parity-Check,GLDPC)码,称为ZS-GLDPC码.该码具有线性编码复杂度,可采用和积译码算法实现迭代译码,其译码复杂度低于以汉明码为分量码的GLDPC码.在均匀交织器的前...     

新型级联Polar码设计

Polar码是一种新型高效的信道编码技术,被确定为5G增强移动宽带场景控制信道的编码方案。本文提出一种循环冗余校验(Cyclic Redundancy Check, CRC)码、奇偶校验(Parity Check, PC)码与Polar码级联方案,其中CRC码、PC码作为外码,Polar码作为内码。与CRC辅助的Polar码方案相比,新型级联Polar码在译码的过程中利用PC比特辅助路径度量值进行译码路径的修剪,用以保证路径选择的可靠性,从而提高了其纠错性能,由于PC操作简单,在复杂度上没有明显增加。仿真结果表明:新型级联Polar码具有优异的性能,当误码率为10-6,码长为512,码率为1/3时,新型级联Polar码与CRC辅助的Polar码相比大约有0.12 dB的增益。      

光码分多址(OCDMA)通信系统中的地址码

介绍了光码分多址 (OCDMA)系统中的地址码 ,包括光正交码 (OOC)、素数码 (PC)、扩展的素数码 (EPC)、二次同余码 (QC)、扩展的二次同余码 (EQC)和混合码 (HC)等。同时分析了它们各自的性能。     

Shortened finite geometry codes (Corresp.)

A method of shortening finite analytic geometry codes, projective-geometry (PG) codes, Euclidean-geometry (EG) codes, and 2-fold EG codes is presented. The shortened codes preserve the feature of being majority-logic decodable and they have the same error-correcting capability as the original codes. Combinatorial expressions for the parity-check symbols of the shortened codes are derived.     

A class of high-speed decodable burst-correcting codes (Corresp.)

A class of high-speed decodable burst-correcting codes is presented. This class of codes is obtained by modifying burst-correcting convolutional codes into block codes and does not require any cyclic shifts in the decoding process. With the appropriate choices of parameters, the codes can approximate minimum-redundancy codes. The high-speed decodability is expected to make these codes suitable for application to computer systems.     

Projective Reed-Muller codes

A class of codes in the Reed-Muller family, the projective Reed-Muller codes (PRM codes), is studied. The author defines the PRM codes of all orders and discusses their relation to polynomial codes. The exact parameters of PRM codes are given. The duals are characterized, and, in parallel to the classical works on generalized Reed-Muller codes, the cyclic properties are studied. Tables over parameters of the codes are given     

Consta-Abelian polyadic codes

In this correspondence, the class of polyadic codes is generalized to the class of consta-Abelian polyadic codes, which, in particular, includes the class of constacyclic polyadic codes. Properties such as the equivalence of polyadic codes and the mth-root lower bound for the minimum weight of a subcode of certain types of polyadic codes are preserved in the consta-Abelian case. Sufficient conditions for the existence of this class of codes are established. For the special case of constacyclic codes, the characterization of negacyclic self-dual codes of length coprime to the characteristic of the field in terms of negacyclic duadic codes is also given     

QPSK block-modulation codes for unequal error protection

Unequal error protection (UEP) codes find applications in broadcast channels, as well as in other digital communication systems, where messages have different degrees of importance. Binary linear UEP (LUEP) codes combined with a Gray mapped QPSK signal set are used to obtain new efficient QPSK block-modulation codes for unequal error protection. Several examples of QPSK modulation codes that have the same minimum squared Euclidean distance as the best QPSK modulation codes, of the same rate and length, are given. In the new constructions of QPSK block-modulation codes, even-length binary LUEP codes are used. Good even-length binary LUEP codes are obtained when shorter binary linear codes are combined using either the well-known |u¯|u¯+v¯|-construction or the so-called construction X. Both constructions have the advantage of resulting in optimal or near-optimal binary LUEP codes of short to moderate lengths, using very simple linear codes, and may be used as constituent codes in the new constructions. LUEP codes lend themselves quite naturally to multistage decoding up to their minimum distance, using the decoding of component subcodes. A new suboptimal two-stage soft-decision decoding of LUEP codes is presented and its application to QPSK block-modulation codes for UEP illustrated     

Sample-based classification procedures related to empiric distributions

We present new classes of binary codes that are constructed on the basis of concatenated codes and product codes. We discuss the random-error-correction capabilities of these codes. Some examples of the codes for the correction of random errors are given which have at least as many codewords as the best codes previously known (to the authors) with the same minimum distance and same number of check symbols. The burst-error-correction capabilities of the codes are also discussed. Several examples of the codes for the correction of both random errors and burst errors are given. A decoding algorithm for the codes is also described.