The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code

An extremal self-dual doubly-even binary (n,k,d) code has a minimum weight d=4/spl lfloor/n/24/spl rfloor/+4. Of such codes with length divisible by 24, the Golay code is the only (24,12,8) code, the extended quadratic residue code is the only known (48,24,12) code, and there is no known (72,36,16) code. One may partition the search for a (48,24,12) self-dual doubly-even code into three cases. A previous search assuming one of the cases found only the extended quadratic residue code. We examine the remaining two cases. Separate searches assuming each of the remaining cases found no codes and thus the extended quadratic residue code is the only doubly-even self-dual (48,24,12) code.     

Algebraic decoding of the ternary (13,7,5) quadratic residue code

An algebraic decoding algorithm for the ternary (13,7,5) quadratic residue code is presented. This seems to be the first attempt to provide an algebraic decoding algorithm for a quadratic residue code over a nonbinary field     

Algebraic decoding of the (32, 16, 8) quadratic residue code

An algebraic decoding algorithm for the 1/2-rate (32, 16, 8) quadratic residue (QR) code is found. The key idea of this algorithm is to find the error locator polynomial by a systematic use of the Newton identities associated with the code syndromes. The techniques developed extend the algebraic decoding algorithm found recently for the (32, 16, 8) QR code. It is expected that the algebraic approach developed here and by M. Elia (1987) applies also to longer QR codes and other BCH-type codes that are not fully decoded by the standard BCH decoding algorithm     

The algebraic decoding of the (41, 21, 9) quadratic residue code

A new algebraic approach for decoding the quadratic residue (QR) codes, in particular the (41, 21, 9) QR code, is presented. The key ideas behind this decoding technique are a systematic application of the Sylvester resultant method to the Newton identities associated with the syndromes to find the error-locator polynomial, and next a method for determining error locations by solving certain quadratic, cubic, and quartic equations over GF(2^m) in a new way which uses Zech's logarithms for the arithmetic. The logarithms developed for Zech's logarithms save a substantial amount of computer memory by storing only a table of Zech's logarithms. These algorithms are suitable for implementation in a programmable microprocessor or special-purpose VLSI chip. It is expected that the algebraic methods developed can apply generally to other codes such as the BCH and Reed-Solomon codes     

On high-speed decoding of the (23,12,7) Golay code

An algebraic decoding method for triple-error-correcting binary BCH codes applicable to complete decoding of the (23,12,7) Golay code has been proved by M. Elia (see ibid., vol.IT-33, p.150-1, 1987). A modified step-by-step complete decoding algorithm of this Golay code is introduced which needs fewer shift operations than Kasami's error-trapping decoder. Based on the algorithm, a high-speed hardware decoder of this code is proposed     

Fast ML decoding algorithm for the Nordstrom - Robinson code (Corresp.)

The extended Nordstrom-Robinson code is an optimum nonlinear double-error-correcting code of length16with considerable practical importance. This code is also useful as a rate1/2vector quantizer for random waveforms such as speech linear-predictive-coding (LPC) residual. A fast decoding algorithm is described for maximum likelihood decoding (or nearest neighbor search in the squared-error sense) with304additions and128comparisons.     

Algebraic decoding of (103, 52, 19) and (113, 57, 15) quadratic residue codes

In this paper, two algebraic decoders for the (103, 52, 19) and (113, 57, 15) quadratic residue codes, which have lengths greater than 100, are presented. The results have been verified by software simulation that programs in C++ language have been executed to check possible error patterns of both quadratic residue codes.     

Universal decoding algorithm for binary quadratic residue codes using syndrome-weight determination

A novel decoding scheme, called syndrome-weight determination, was proposed by Chang et al. in 2008 for the Golay code, or the (23, 12, 7) quadratic residue code. This method is not only very simple in principle but also suitable for parallel hardware design. Presented is a modified version for any binary quadratic residue codes which has been developed. Because of its regular property, the proposed decoder is suitable for both software design and hardware development.     

Algebraic decoding of the (23,12,7) Golay code (Corresp.)

It is shown that the algebraic method for decoding three-error-correcting BCH codes is also applicable to complete decoding of the(23,12,7)Golay code.     

(24,16)循环码编译码方法研究

在理论分析循环码编码和译码基本原理的基础上,提出了基于单片机系统的(24,16)循环码软件实现编码、译码的方案.仿真结果表明(24,16)循环码能有效地克服来自通讯信道的干扰,保证数据通信的可靠及系统的稳定,使误码率大幅度降低.本论文对(24,16)循环码的研究结果表明,可以有效地降低错误概率和提高系统的吞吐量,实现纠...     

On the minimum distance of some quadratic residue codes (Corresp.)

We find the minimum distances of the binary(113, 57), and ternary(37, 19), (61, 31), (71, 36), and(73, 37)quadratic residue codes and the corresponding extended codes. These distances are15, 10, 11, 17, and17, respectively, for the nonextended codes and are increased by one for the respective extended codes. We also characterize the minimum weight codewords for the(113, 57)binary code and its extended counterpart.     

No binary quadratic residue code of length 8m-1 is quasi-perfect

The class of binary quadratic residue (QR) codes of length n=8m-1 contains two perfect codes. These are the (7,4,3) Hamming code and the (23,12,7) Golay code. However, it is proved in the present paper that there are no quasi-perfect QR codes of length 8m-1. Finally, this result is generalized to all binary self-dual codes of length N>72     

On the binary quadratic residue system with noncoprime moduli

The residue number system (RNS) appropriate for implementing fast digital signal processors since it can support parallel, carry-free, high-speed arithmetic. A development in residue arithmetic is the quadratic residue number system (QRNS), which can perform complex multiplications with only two integer multiplications instead of four. An RNS/QRNS is defined by a set of relatively prime integers, called the moduli set, where the choice of this set is one of the most important design considerations for RNS/QRNS systems. In order to maintain simple QRNS arithmetic, moduli sets with numbers of forms 2ⁿ+1 (n is even) have been considered. An efficient such set is the three-moduli set (2^2k-2+1.2^2k+1.2^2k+2+1) for odd k. However, if large dynamic ranges are desirable, QRNS systems with more than three relatively prime moduli must be considered. It is shown that if a QRNS set consists of more than four relatively prime moduli of forms 2ⁿ+1, the moduli selection process becomes inflexible and the arithmetic gets very unbalanced. The above problem can be solved if nonrelatively prime moduli are used. New multimoduli QRNS systems are presented that are based on nonrelatively prime moduli of forms 2ⁿ +1 (n even). The new systems allow flexible moduli selection process, very balanced arithmetic, and are appropriate for large dynamic ranges. For a given dynamic range, these new systems exhibit better speed performance than that of the three-moduli QRNS system     

On minimal decoding sets for the extended binary Golay code

Minimal decoding sets consisting of fourteen permutations have been found for the (24,12,8) extended binary Golay code. It is shown that by properly sequencing the permutations of one such set, the average number of permutations required to decode a random received word can be minimized     

部分正交空时块编码(POSTBC)及其解码算法

提出一种三发射天线的部分正交空时编码(POSTBC)结构.发送天线被分成两组,第一组采用Alamouti STBC编码,第二组则直接采用QPSK调制 .和Alamouti STBC 分组码相比它的优点是不需要偶数根发射天线. 由于各组发送的数据流相互独立,针对这一系统框架,采用了最优排序串行干扰消除(OSIC)译码算法和一种新的译码算法.计算机仿真结果表明,这种空时编码(MIMO)结构在等数据率的情况下能获得比相应的V-BLAST系统更优的性能.     

Algebraic decoding of (71, 36, 11), (79, 40, 15), and (97, 49, 15) quadratic residue codes

Recently, a new algebraic decoding algorithm for quadratic residue (QR) codes was proposed by Truong et al. Using that decoding scheme, we now develop three decoders for the QR codes with parameters (71, 36, 11), (79, 40, 15), and (97, 49, 15), which have not been decoded before. To confirm our results, an exhaustive computer simulation has been executed successfully.     

On general linear block code decoding using the sum-product iterative decoder

The sum-product iterative decoder, conventionally used for low-density parity-check (LDPC) codes, hold promise as a decoder for general linear block code decoding. However, the promise is only partly fulfilled because, as we show experimentally, the decoder performance degrades rapidly as a function of parity check matrix weight. Even in the case of decoder failure, however, we demonstrate that there is information present in the decoder output probabilities that can still help with the decoding problem.     

Soft-decision decoding of the teletext character code

This letter proposes a soft-decision decoding algorithm as a means of improving the displayed-character error rate of teletext transmissions by simple modifications to the decoder only. The expected improvement is theoretically assessed, performance curves are given and implementation of the scheme is discussed.     

Fast maximum-likelihood decoding of the golden code

Because each golden code codeword conveys four information symbols from an M-ary QAM alphabet, the complexity of an exhaustive-search decoder is proportional to M⁴. In this paper we prove that the golden code is fast-decodable, meaning that maximum-likelihood decoding is possible with a worst-case complexity proportional to only M^2.5. The golden code retains its fast-decodable property regardless of whether the channel varies with time. We also present an efficient implementation of a fast maximum-likelihood decoder that exhibits a low average complexity.