The Partition Weight Enumerator of MDS Codes and its Applications

A closed form formula of the partition weight enumerator of maximum distance separable (MDS) codes is derived for an arbitrary number of partitions. Using this result, some properties of MDS codes are discussed. The results are extended for the average binary image of MDS codes in finite fields of characteristic two. As an application, we study the multiuser error probability of Reed Solomon codes.Comment: This is a five page conference version of the paper which was accepted by ISIT 2005. For more information, please contact the author

A Method to determine Partial Weight Enumerator for Linear Block Codes

In this paper we present a fast and efficient method to find partial weight enumerator (PWE) for binary linear block codes by using the error impulse technique and Monte Carlo method. This PWE can be used to compute an upper bound of the error probability for the soft decision maximum likelihood decoder (MLD). As application of this method we give partial weight enumerators and analytical performances of the BCH(130,66), BCH(103,47) and BCH(111,55) shortened codes; the first code is obtained by shortening the binary primitive BCH (255,191,17) code and the two other codes are obtained by shortening the binary primitive BCH(127,71,19) code. The weight distributions of these three codes are unknown at our knowledge.Comment: Computer Engineering and Intelligent Systems Vol 3, No.11, 201

Multi-dimensional direct-sequence spread spectrum multiple-access communication with adaptive channel coding

During the race towards the4th generation (4G) cellular-based digital communication systems, a growth in the demand for high capacity, multi-media capable, improved Quality-of-Service (QoS) mobile communication systems have caused the developing mobile communications world to turn towards betterMultiple Access (MA) techniques, like Code Division Multiple Access (CDMA) [5]. The demand for higher throughput and better QoS in future 4G systems have also given rise to a scheme that is becoming ever more popular for use in these so-called ‘bandwidth-on-demand’ systems. This scheme is known as adaptive channel coding, and gives a system the ability to firstly sense changes in conditions, and secondly, to adapt to these changes, exploiting the fact that under good channel conditions, a very simple or even no channel coding scheme can be used for Forward Error Correction(FEC). This will ultimately result in better system throughput utilization. One such scheme, known as incremental redundancy, is already implemented in the Enhanced Data Rates for GSM Evolution (EDGE) standard. This study presents an extensive simulation study of a Multi-User (MU), adaptive channel coded Direct Sequence Spread Spectrum Multiple Access (DS/SSMA) communication system. This study firstly presents and utilizes a complex Base Band(BB) DS/SSMA transmitter model, aimed at user data diversity [6] in order to realize the MU input data to the system. This transmitter employs sophisticated double-sideband (DSB)Constant-Envelope Linearly Interpolated Root-of-Unity (CE-LI-RU) filtered General Chirp-Like (GCL) sequences [34, 37, 38] to band limit and spread user data. It then utilizes a fully user-definable, complex Multipath Fading Channel Simulator(MFCS), first presented by Staphorst [3], which is capable of reproducing all of the physical attributes of realistic mobile fading channels. Next, this study presents a matching DS/SSMA receiver structure that aims to optimally recover user data from the channel, ensuring the achievement of data diversity. In order to provide the basic channel coding functionality needed by the system of this study, three simple, but well-known channel coding schemes are investigated and employed. These are: binary Hamming (7,4,3) block code, (15,7,5) binary Bose-Chadhuri-Hocquenghem (BCH) block code and a rate 1/3 <i.Non-Systematic (NS) binary convolutional code [6]. The first step towards the realization of any adaptive channel coded system is the ability to measure channel conditions as fast as possible, without the loss of accuracy or inclusion of known data. In 1965, Gooding presented a paper in which he described a technique that measures communication conditions at the receiving end of a system through a device called a Performance Monitoring Unit (PMU) [12, 13]. This device accelerates the system’sBit Error Rate (BER) to a so-called Pseudo Error Rate(PER) through a process known as threshold modification. It then uses a simple PER extrapolation algorithm to estimate the system’s true BER with moderate accuracy and without the need for known data. This study extends the work of Gooding by applying his technique to the DS/SSMA system that utilizes a generic Soft-Output Viterbi Algorithm(SOVA) decoder [39] structure for the trellis decoding of the binary linear block codes [3, 41-50], as well as binary convolutional codes mentioned, over realistic MU frequency selective channel conditions. This application will grant the system the ability to sense changes in communication conditions through real-time BER measurement and, ultimately, to adapt to these changes by switching to different channel codes. Because no previous literature exists on this application, this work is considered novel. Extensive simulation results also investigate the linearity of the PER vs. modified threshold relationship for uncoded, as well as all coded cases. These simulations are all done for single, as well as multiple user systems. This study also provides extensive simulation results that investigate the calculation accuracy and speed advantages that Gooding’s technique possesses over that of the classic Monte-Carlo technique for BER estimation. These simulations also consider uncoded and coded cases, as well as single and multiple users. Finally, this study investigates the experimental real-time performance of the fully functional MU, adaptive coded, DS/SSMA communication system over varying channel conditions. During this part of the study, the channel conditions are varied over time, and the system’s adaptation (channel code switching) performance is observed through a real-time observation of the system’s estimated BER. This study also extends into cases with multiple system users. Since the adaptive coded system of this study does not require known data sequences (training sequences), inclusion of Gooding’s technique for real-time BER estimation through threshold modification and PER extrapolation in future 4G adaptive systems will enable better Quality-of-Service (QoS) management without sacrificing throughput. Furthermore, this study proves that when Gooding’s technique is applied to a coded system with a soft-output, it can be an effective technique for QoS monitoring, and should be considered in 4G systems of the future.Dissertation (MEng (Computer Engineering))--University of Pretoria, 2007.Electrical, Electronic and Computer EngineeringMEngunrestricte

Topologically Driven Methods for Construction Of Multi-Edge Type (Multigraph with nodes puncturing) Quasi-Cyclic Low-density Parity-check Codes for Wireless Channel, WDM Long-Haul and Archival Holographic Memory

In this Phd thesis discusses modern methods for constructing MET QC-LDPC codes with a given error correction ("waterfall, error-floor") and complexity (parallelism level according circulant size plus scheduler orthogonality of checks) profiles: 1. weight enumerators optimization, protograph construction using Density Evolution, MI (P/Exit-chart) and it approximation: Gaussian Approximation, Reciprocal-channel approximation and etc; 2. Covariance evolution and it approximation; 3. Lifting methods for QC codes construction:PEG, Guest-and-Test, Hill-Climbing with girth, EMD, ACE optimization; 4. Upper and lower bounds on code distance estimation and its parallel implementation using CPU/GPU; 5. Brouwer-Zimmerman and Number Geometry code distance estimation methods; 6. Importance Sampling for error-floor estimation; 7. Length and rate adaption methods for QC codes based on cyclic group decomposition; 8. Methods for interaction screening which allow to improve performance (decorrelate variables) under BP and it's approximation. We proposed several state-of-the-art methods: Simulated Annealing lifting for MET QC-LDPC codes construction; fast EMD and code distance estimation; floor scale modular lifting for lenght adaption; fast finite-length covariance evolution rate penalty from threshold for code construction and it hardware friendly compression for fast decoder's LLRs unbiasing due SNR's estimation error. We found topology reason's of efficient of such methods using topology thickening (homotopy of continuous and discrete curvature) under matched metric space which allow to generalize this idea to a class of nonlinear codes for Signal Processing and Machine Learning. Using the proposed algorithms several generations of WDM Long-Haul error-correction codes were built. It was applied for "5G eMBB" 3GPP TS38.212 and other applications like Flash storage, Compressed sensing measurement matrix.Comment: Phd Thesis, 176 pages, in Russian, 62 pictures, 13 tables, 5 appendix including links to binary and source code