Golay Code Research Articles

A New Construction for the Extended Binary Golay Code

We give a new construction of the extended binary Golay code. The construction is carried out by taking the Gray image of a self-dual linear code over the ring R=F2+ uF2+ vF2+ uvF2 of length 6 and size 2 12 . Writing a typical generating matrix of the form (I3jA), with A being a 3� 3 matrix over R, and finding some dependencies among the entries of A, we are able to set a general form for the generating matrices of self-dual codes of length 6. Using some special properties of elements of R, we end up with a family of generating matrices all of which give us the extended binary Golay code. We also prove the minimum distance property analytically.

Performance Analysis of Modulation and Coding to Maximize the Lifetime of Wireless Sensor Network

A wireless sensor network (WSN) consists of several sensor nodes to monitor physical or environmental conditions. The development of WSNs is motivated by several applications such as military surveillance, industrial and consumer applications. In this paper transmission mechanism of a sensor node is analyzed with modulation schemes and error control codes. It furthermore compared based on channel condition in a sensor network. The modulation schemes considered for this work including 16-PSK, 16-PAM, 16-QAM and 16-FSK along with convolutional, Golay and RS codes under both AWGN (Additive White Gaussian Noise) and Rayleigh fading channels. To maximize the lifetime of WSN, appropriate combination of modulation scheme with error control codes is chosen for sensor data transmission. The result shows that 16-FSK with Golay codes in AWGN and 16-QAM with Golay codes in Rayleigh channel is more energy efficient than other combination of modulation and coding techniques for energy efficient sensor data transmission

On the bounds and achievability about the ODPC of $$\mathcal{GRM }(2,m)^*$$ GRM ( 2 , m ) ∗ over prime fields for increasing message length

The optimum distance profiles of linear block codes were studied for increasing or decreasing message length while keeping the minimum distances as large as possible, especially for Golay codes and the second-order Reed---Muller codes, etc. Cyclic codes have more efficient encoding and decoding algorithms. In this paper, we investigate the optimum distance profiles with respect to the cyclic subcode chains (ODPCs) of the punctured generalized second-order Reed---Muller codes $$\mathcal{GRM }(2,m)^*$$ GRM ( 2 , m ) ? which were applied in Power Control in OFDM Modulations, in channels with synchronization, and so on. For this, two standards are considered in the inverse dictionary order, i.e., for increasing message length. Four lower bounds and upper bounds on ODPC are presented, where the lower bounds almost achieve the corresponding upper bounds in some sense. The discussions are over nonbinary prime fields.

Efficient decoding of the (23, 12, 7) Golay code up to five errors

A novel and efficient algorithm for decoding the (23, 12, 7) Golay code with the five-error correcting capability is developed. Core to the key idea is innovatively utilizing the relationship of syndromes among error patterns, which is derived from the property of perfect code. Consequently, two methods of fast searching candidate codewords are proposed, both of which only perform the algebraic hard decision decoder once, rather than iterative decoding of Chase-like algorithm. With the set of the possible codewords, the most likely one is chosen as an output codeword based on the correlation metric. In comparison to Chase-2 algorithm, simulation results over the AWGN channel reveal the decoding times of the proposed algorithm equipped with codeword matching and syndrome-group search are reduced by 25% and 75%, respectively. In terms of the percentage of correct decoding, it turns out that the proposed algorithm outperforms Chase-2 algorithm, especially in the error-prone transmission environment. These favorable results demonstrate that the new algorithm is beneficial to implement in practice.

The use of Walsh code in modulating the pump light of high spatial resolution phase-shift-pulse Brillouin optical time domain analysis with non-return-to-zero pulses

We have previously reported phase modulation of pump light of phase-shift pulse Brillouin optical time domain analysis with non-return-to-zero (NRZ)-formatted Golay codes. However, the Brillouin signal has been distorted when the duration of the coded pulse used exceeds the time constant of the amplitude in the acoustic wave that causes stimulated Brillouin scattering. Therefore, we propose pump modulation with Walsh codes to overcome this problem. We have shown that the use of Walsh codes in the NRZ-formatted pump induces little error in the spatial resolution, contrary to the previous results for Golay codes. The maximum signal-to-noise ratio improvement for 8-bit Walsh codes recorded was 3 dB and the spatial resolution obtained was 10 cm, being consistent with analytical and numerical calculations.

Bipolar optical pulse coding for performance enhancement in BOTDA sensors

A pump signal based on bipolar pulse coding and single-sideband suppressed-carried (SSB-SC) modulation is proposed for Brillouin optical time-domain analysis (BOTDA) sensors. Making a sequential use of the Brillouin gain and loss spectra, the technique is experimentally validated using bipolar complementary-correlation Golay codes along a 100 km-long fiber and 2 m spatial resolution, fully resolving a 2 m hot-spot at the end of the sensing fiber with no distortion introduced by the decoding algorithm. Experimental results, in good agreement with the theory, indicate that bipolar Golay codes provide a higher signal-to-noise ratio enhancement and stronger robustness to pump depletion in comparison to optimum unipolar pulse codes known for BOTDA sensing.

Automorphisms of Order $2p$ in Binary Self-Dual Extremal Codes of Length a Multiple of 24

Let C be a binary self-dual code with an automorphism g of order 2p, where p is an odd prime, such that gp is a fixed point free involution. If C is extremal of length a multiple of 24, all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code C with coding theoretical ones of the subcode C(gp) which consists of the set of fixed points of gp, we prove that C is a projective F2〈g 〉-module if and only if a natural projection of C(gp) is a self-dual code. We then discuss easy-to-handle criteria to decide if C is projective or not. As an application, we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58.

High-Speed Decoding of the Binary Golay Code

Recently, some table-lookup decoding algorithms (TLDAs) have been used to correct the binary Golay code. Thispaper proposes an efficient high-speed TLDA called message-syndrome decoding algorithm (MSDA) by using themessage syndrome to correct the binary systematic Golay code. The proposed MSDA is based on the novelmessage-syndrome lookup table (MSLT). The MSLT merely consists of 12 candidate syndromes and its memory sizeis significantly smaller than other existing lookup tables. Computer software simulation results show that the decodingspeed of the proposed MSDA is superior to other proposed TLDAs.

Reduction of Side Lobes by Using Complementary Codes for Radar Application

The analysis of new types of Complementary direct sequence complex signals which have synthesized with well - know code sequences like Barker, Walsh, Golay, and complementary codes. Build on the autocorrelation function (ACF) and ambiguity function (AF) of signals, the numerical method estimates the volume of side lobes separately for each signal. The results obtained show that the signals, which have a low volume of side lobes, means of approximately zero in by using complementary codes with compare to different codes.

Combinatorics associated with inflections and bitangents of plane quartics

After a preliminary survey and a description of some small Steiner systems from the standpoint of the theory of invariants of binary forms, we construct a binary Golay code (of length 24) using ideas from J. Grassmann's thesis of 1875. One of our tools is a pair of disjoint Fano planes. Another application of such pairs and properties of plane quartics is a construction of a new block design on 28 objects. This block design is a part of a dissection of the set of 288 Aronhold sevens. The dissection distributes the Aronhold sevens into 8 disjoint block designs of this type.

Soft decoding of the (23, 12, 7) Golay code: simplified algorithm and hardware implementation

A simplified soft decoding algorithm for the (23, 12, 7) Golay code that can correct up to six errors is proposed in this study. The emblematic probability value of this algorithm derived from Reed is shown to be the same as the one given by Chase, if a joint estimate of the noise power and the received bit amplitude are carried out at the output of the matched filter in a Costas-loop receiver. Simulation results show that the coding gain obtained by the simplified algorithm is nearly the same as that achieved by Chase's algorithm. Based on the simplified algorithm, this study investigates an efficient hardware implementation on field programmer gate array platform, and presents a detailed architecture of the soft decoder. Experimental result indicates that the proposed algorithm uses less hardware complexity and has high-throughput rate, therefore it is naturally suitable for very large scale integration (VLSI) implementation.

A base-sequence-modulated golay code improves the excitation and measurement of ultrasonic guided waves in long bones

Researchers are interested in using ultrasonic guided waves (GWs) to assess long bones. However, GWs suffer high attenuation when they propagate in long bones, resulting in a low SNR. To overcome this limitation, this paper introduces a base-sequence-modulated Golay code (BSGC) to produce larger amplitude and improve the SNR in the ultrasound evaluation of long bones. A 16-bit Golay code was used for excitation in computer simulation. The decoded GWs and the traditional GWs, which were generated by a single pulse, agreed well after decoding the received signals, and the SNR was improved by 26.12 dB. In the experiments using bovine bones, the BSGC excitation produced the amplitudes which were at least 237 times greater than those produced by a single pulse excitation. The BSGC excitation also allowed the GWs to be received over a longer distance between two transducers. The results suggest the BSGC excitation has the potential to measure GWs and assess long bones.

Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code

In fault-tolerant quantum computing schemes, the overhead is often dominated by the cost of preparing codewords reliably. This cost generally increases quadratically with the block size of the underlying quantum error-correcting code. In consequence, large codes that are otherwise very efficient have found limited fault-tolerance applications. Fault-tolerant preparation circuits therefore are an important target for optimization. We study the Golay code, a $23$-qubit quantum error-correcting code that protects the logical qubit to a distance of seven. In simulations, even using a na{\"i}ve ancilla preparation procedure, the Golay code is competitive with other codes both in terms of overhead and the tolerable noise threshold. We provide two simplified circuits for fault-tolerant preparation of Golay code-encoded ancillas. The new circuits minimize error propagation, reducing the overhead by roughly a factor of four compared to standard encoding circuits. By adapting the malignant set counting technique to depolarizing noise, we further prove a threshold above $\threshOverlap$ noise per gate.

Slow and fast ultrasonic wave detection improvement in human trabecular bones using Golay code modulation

The identification of fast and slow waves propagating through trabecular bone is a challenging task due to temporal wave overlap combined with the high attenuation of the fast wave in the presence of noise. However, it can provide valuable information about bone integrity and become a means for monitoring osteoporosis. The objective of this work is to apply different coded excitation methods for this purpose. The results for single-sine cycle pulse, Golay code, and chirp excitations are compared. It is shown that Golay code is superior to the other techniques due to its signal enhancement while exhibiting excellent resolution without the ambiguity of sidelobes.

Range Sidelobes Suppression in Pulse-Compression Radar using Golay Pairs: Some Basic Limitations for Complex Targets

This study deals with a specific aspect of the well-known problem of the variation of the radar cross section of complex targets with the transmitted frequency. In particular, it considers pairs of modified Golay codes and demonstrates that-even with a modest variation of the transmitted frequency-the square complementary properties of radar returns are lost when targets are complex (i.e., made up by some point scatterers). This fact causes the range sidelobes to increase very much. A partial remedial measure to this drawback may be the simultaneous transmission of both codes by an array antenna at the same frequency.

Generalization of all similar constructions of extended binary Golay code

Generalization of all similar constructions of extended binary Golay code

Golay code modulation in low-power laser-ultrasound

The current work presents a correlation-based detection technique with application in modulated laser-ultrasonics. 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 ∼1W. In the experiments a plate with a thickness of 50μm 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 27=128, 29=512, and 211=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.

On Pseudocodewords and Decision Regions of Linear Programming Decoding of HDPC Codes

In this paper we explore the decision regions of Linear Programming (LP) decoding. We compare the decision regions of an LP decoder, a Belief Propagation (BP) decoder and the optimal Maximum Likelihood (ML) decoder. We study the effect of minimal-weight pseudocodewords on LP decoding. We present global optimization as a method for finding the minimal pseudoweight of a given code as well as the number of minimal-weight generators. We present a complete pseudoweight distribution for the [24, 12, 8] extended Golay code, and provide justifications of why the pseudoweight distribution alone cannot be used for obtaining a tight upper bound on the error probability.

Design and performance of Huffman sequences in medical ultrasound coded excitation

This paper deals with coded-excitation techniques for ultrasound medical echography. Specifically, linear Huffman coding is proposed as an alternative approach to other widely established techniques, such as complementary Golay coding and linear frequency modulation. The code design is guided by an optimization procedure that boosts the signal-to-noise ratio gain (GSNR) and, interestingly, also makes the code robust in pulsed-Doppler applications. The paper capitalizes on a thorough analytical model that can be used to design any linear coded-excitation system. This model highlights that the performance in frequency-dependent attenuating media mostly depends on the pulse-shaping waveform when the codes are characterized by almost ideal (i.e., Kronecker delta) autocorrelation. In this framework, different pulse shapers and different code lengths are considered to identify coded signals that optimize the contrast resolution at the output of the receiver pulse compression. Computer simulations confirm that the proposed Huffman codes are particularly effective, and that there are scenarios in which they may be preferable to the other established approaches, both in attenuating and non-attenuating media. Specifically, for a single scatterer at 150 mm in a 0.7-dB/(MHz·cm) attenuating medium, the proposed Huffman design achieves a main-to-side lobe ratio (MSR) equal to 65 dB, whereas tapered linear frequency modulation and classical complementary Golay codes achieve 35 and 45 dB, respectively.

Low Complexity Distributed Video Coding with Golay Codes

In this paper we present a low complexity video coder working on the principle of distributed source coding that combines the principle of channel coding with source coding. In this work the encoder complexity is shifted to the decoder to support uplink friendly video applications, simultaneously achieving the rate-distortion performance of the conventional predictive coding system. In this work concept of syndrome coding with Golay codes is adopted for compression. The simulation results presented in this paper reveals the superior performance of this distributed video coder over the Intraframe coders in terms of rate distortion performance, simultaneously achieving low complexity when compared to predictive coders.