Golay Code

In this paper, a fast and efficient decoding algorithm for correcting the (23, 12, 7) Golay code up to four errors is presented. The aim of this paper is to develop a fast syndrome-group search method for finding the candidate codewords by utilizing the property that the syndromes of the weight-4 error patterns are identical to that of the weight-3 error pattern. When the set of the candidate codewords is constructed, the most likely one is determined by assessing the corresponding correlation metrics. The well-known Chase-II decoder, which needs to perform the hard-decision decoder multiple times, acts as a comparison basis. Simulation results over the additive white Gaussian noise channel show that the decoding complexity of the proposed method is averagely reduced by at least 86% in terms of the decoding time. Furthermore, the successful decoding percentage of the new decoder in the case of four errors is always superior to Chase-II decoder. At the signal-tonoise ratio of 0 dB, the proposed algorithm still can correct up to 97.40% weight-4 error patterns. The overall bit error rate performance of the proposed decoder is close to that of Chase-II decoder. It implies that the new decoder is beneficial to the practical implementation.

New multilevel constructions of the Golay code and the Leech lattice are presented. These are derived from the 'hryn construction and the "holy construction" with the octacode as the glue code. Further, we show that the "holy construction" of the Leech lattice with the octacode as the glue code is essentially different from the permuted Turyn construction, although both constructions rely on the octacode. Turyn construction is based on an "odd" type of the octacode, whereas any type of the octacode can be used in the "holy construction." Moreover, the multilevel representation of the "holy construction" leads to a novel lattice partition chain. Based on these structures, we derive new bounded-distance decoders for the Golay code and the Leech lattice whose effective error coefficient is smaller than that of any previously known bounded-distance decoder. We provide a general theorem for computing the effective error coefficient of coset decoding with bounded distance decoding for the subcode.

Channel coding is commonly incorporated to obtain sufficient reception quality in wireless mobile communications transceiver to counter channel degradation due to inter-symbol interference, multipath dispersion, and thermal noise induced by electronic circuit devices. High speed and high throughput hardware for encoder and decoder could be useful in communication field. Due to the channel achieving property, the GOLAY code has become one of the most favorable error-correcting codes. In this paper, a new algorithm has been proposed for CRC based encoding scheme, which devoid of any linear feedback shift registers (LFSR). In this architecture, our work is to design a GOLAY code based encoder and decoder architecture using CRC processing technique. The other approach is to design a Bloom filter with hamming distance based fast error detection and correction methodology and this work is to improve the secure data transmission. The bloom filter system is to identify the error bit location using the redundant bits add process and to correct the error for XOR based distance calculation process. The Bloom filter architecture is used to set the hash value for allocated transmitted data sequence and to improve the fault identification methodology. This method is to optimize the decoder structure and effectively identify the error location, then to correct to error using bit reverser logic process.

The current work presents a correlation-based detection technique with application in modulated laserultrasonics. In standard use of coded sequences the impulse response of a system is recovered in the time domain with improved signal to noise ratio (SNR). The presented method is an extension of this technique, where the response to a chirped waveform is restored with improved SNR; hence, the response is in a well-defined frequency range. To achieve this goal the chirped waveforms are modulated by Golay codes. It will be shown that the response to this bandlimited carrier waveform can be recovered in the time domain with improved signal to noise ratio using a cross-correlation technique. Improvement in the SNR is discussed analytically and it is shown that this improvement is proportional to the square root of the length of the applied sequences. Experimental applications in laser-ultrasound are shown using modulated laser diodes as excitation sources with an output power of $1 W. In the experiments a plate with a thickness of 50 lm is investigated using Lamb waves in the MHz range to confirm the predicted improvement in the SNR. Golay codes with three different lengths were used with 7, 9 and 11 bits resulting in 2 7 = 128, 2 9 = 512, and 2 11 = 2048 repetitions in an individual signal, respectively. The predicted improvements of 2 in the SNR between the 7 and 9 bits, and between the 9 and 11 bits waveforms, respectively, were well approximated by the experimentally obtained values of 1.83 and 2.17. As Lamb wave dispersion curves can be used for the characterization of plates or layered samples by inverse problems, it is also shown that by using multiple measurement points the recovered waveforms can be utilized in the evaluation of the dispersion relation.

We study a construction of the Mathieu group M12 using a game reminiscent of Loyd's \15-puzzle". The elements of M12 are realized as permu- tations on 12 of the 13 points of the nite projective plane of order 3. There is a natural extension to a \pseudogroup" M13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the con- struction is a metric, akin to that induced by a Cayley graph, on both M12 and M13. We develop these results, and extend them to the double covers and automorphism groups of M12 and M13, using the ternary Golay code and 12 12 Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.

The efficiency in bandwidth utilization and resilience to channel-induced fading makes the OFDM a preferred multicarrier modulation scheme. But the sensitivity of OFDM to frequency and time synchronization errors needs due consideration to avoid system degradation. The purpose of time synchronization is to compensate time offsets and identify the correct starting point of the OFDM frame and symbol at the demodulator. This paper analyzes the offset estimation capability of commonly used preamble sequences and proposes a new preamble structure incorporating Golay sequence with improved merit factor. The simulated results show that the proposed method outperforms the classical time offset estimation methods.

code E6 over an Abelian group A4 with four elements is presented. E6 is cyclic, unlike the [6,3,4] hexacode HG over GF(4). However, E 6 and Hfi are isomorphic when the latter is viewed as a group code. Differences and similarities between E6 and H 6 are discussed. A dual code of E6 is presented. Some binary codes, among them the [24.12.8]Golay, are derived with the aid of E6 A related cyclic [1 2 31 code E; is applied to construct the Nordstrom-Robinson code. €6 is the smallest member of a class of [2k, k , 41 cyclic and reversible codes over Aq Another class of cyclic and reversible codes of length 21 + 1 E 2 2 and minimum distance 3 over A4 is also presented.

New number-theoretic transforms are derived from known linear block codes over finite fields. In particular, two new such transforms are built from perfect codes, namely the Hamming number-theoretic transform and the Golay number-theoretic transform. A few properties of these new transforms are presented.

New number-theoretic transforms are derived from known linear block codes over finite fields. In particular, two new such transforms are built from perfect codes, namely the \textit {Hamming number-theoretic transform} and the \textit {Golay number-theoretic transform}. A few properties of these new transforms are presented.

New number-theoretic transforms are derived from known linear block codes over finite fields. In particular, two new such transforms are built from perfect codes, namely the Hamming number-theoretic transform and the Golay number-theoretic transform. A few properties of these new transforms are presented.