Portable random number generators

Computers are deterministic devices, and a computer-generated random number is a contradiction in terms. As a result, computer-generated pseudorandom numbers are fraught with peril for the unwary. We summarize much that is known about the most well-known pseudorandom number generators: congruential generators. We also provide machine-independent programs to implement the generators in any language that has 32-bit signed integers-for example C, C++, and FORTRAN. Based on an extensive search, we provide parameter values better than those previously available.Programming (Mathematics) ; Computers

A novel maximal-length sequence synchronisation network

Spread Spectrum has become a popular digital modulation scheme in recent years. The advantages the scheme offers, at the expense of bandwidth, make it attractive in a multitude of commercial applications. The most common method, and the one of interest in this thesis, of generating Spread Spectrum is multiplying the data waveform by a wideband, digitally generated waveform. This is referred to as Direct Sequence Spread Spectrum. The characteristics of Spread Spectrum systems are determined by the spreading waveform. A common group of spreading waveforms, and the ones dealt with in this text, are the maximal-length sequences. These are a class of pseudorandom waveforms. Their properties include a two valued autocorrelation function with its maximum value at no code-phase offset. This allows for multiple access to a single resource and the suppression of multi-path interference as adjacent codes have little effect on each other. This same property requires that the receiver must accurately align its replica of the spreading waveform to the transmitted waveform in order to despread the received waveform and demodulate the data. Common methods of synchronisation use a two pronged solution. Firstly the correct code phase is determined. This is referred to as code acquisition. Secondly the clocking frequency of the received waveform must be resolved in order to precisely align the two sequences. This is referred to as code tracking. Receivers therefore tend to be complex and expensive. This thesis involved the investigation of two pseudo-noise synchronisation networks proposed by J .G. van de Groenendaal. These networks offered both code acquisition and tracking in a single robust loop. The investigation, done in co-operation with J..G. van de Groenendaal, persued two avenues. Firstly the loops were simulated. This method allows for the easy alteration of system parameters. Valuable insight into the loop dynamics can thus be gained. Secondly the loops were built on the bench. This allows for the practical confirmation of the results of the simulation. Both synchronisation loops were based on variations of the maximal likelihood phase detector. This phase detector is formed by taking the product of the first derivative with respect to time of the receiver's replica of the transmitted waveform and the received waveform. The initial investigation involved calculating the phase information generated by this phase discriminator for a variety of code-phase and frequency offsets. It was found that there were two stable points in the baseband Spread Spectrum search grid, a grid where a cell consists of a certain code-phase and frequency offset. These stable points existed at no frequency offset, which means that the loops should track the input frequency, and a one or no code-phase offset, which means that the loops should acquire either code-phase. A simple model where the novel synchronisation loop's conditions are represented by a 'ball' resting on the baseband Spread Spectrum search grid as expressed in terms of the integrated phase output of the maximal likelihood phase discriminator was developed. In this model the 'ball' will roll around the surface until one of the two stable points is entered. This describes quite accurately the paths the novel synchronisation loop does in fact take through the baseband Spread Spectrum search grid. The first loop is based directly on the maximal likelihood phase detector. The differentiator is thus in the feedback path of the loop. This results in the loop being unstable and parameter sensitive. Moving the differentiator into the input path, as in the second loop, resulted in a more stable loop. This loop therefore offered a complete, simple synchronisation solution. The novel synchronisation loop with the differentiator in the input path was found to operate at signal-to- noise ratios of -2 dB. Improvement of this signal-to-noise ratio does not offer any advantages in a Spread Spectrum environment as the loop needs to work in a coherent system where the radio frequency carrier must be resolved before the receiver's pseudo-noise sequence can be synchronised. A radio frequency carrier cannot be easily resolved at signal-to-noise ratios lower than O dB. The loop was further adapted to operate in the data environment. Under conditions of data modulation the received waveform is randomly inverted by the data. This results in the loop being driven out of lock. The phase discriminator's slope, having locked on a certain polarity, cannot track an input of the opposite polarity. The loop was adapted by including detection circuitry that would monitor the state of the receiver with respect to the incoming data waveform and alter the polarity of the of the discriminator's slope where necessary. During the prototyping of the loop on the bench certain implementations were investigated. These included the signed edge detector, a wideband low noise implementation of a square wave differentiator, and the synchronous oscillator, a form of injection locked oscillator. The loop was shown to achieve synchronisation. The novel synchronisation loop with the differentiator in the input path is thus capable of synchronising two maximal-length sequences in both code-phase and frequency

Pseudorandom Linear Codes are List Decodable to Capacity

We introduce a novel family of expander-based error correcting codes. These codes can be sampled with randomness linear in the block-length, and achieve list-decoding capacity (among other local properties). Our expander-based codes can be made starting from any family of sufficiently low-bias codes, and as a consequence, we give the first construction of a family of algebraic codes that can be sampled with linear randomness and achieve list-decoding capacity. We achieve this by introducing the notion of a pseudorandom puncturing of a code, where we select $n$ indices of a base code $C\subset \mathbb{F}_q^m$ via an expander random walk on a graph on $[m]$. Concretely, whereas a random linear code (i.e. a truly random puncturing of the Hadamard code) requires $O(n^2)$ random bits to sample, we sample a pseudorandom linear code with $O(n)$ random bits. We show that pseudorandom puncturings satisfy several desirable properties exhibited by truly random puncturings. In particular, we extend a result of (Guruswami Mosheiff FOCS 2022) and show that a pseudorandom puncturing of a small-bias code satisfies the same local properties as a random linear code with high probability. As a further application of our techniques, we also show that pseudorandom puncturings of Reed Solomon codes are list-recoverable beyond the Johnson bound, extending a result of (Lund Potukuchi RANDOM 2020). We do this by instead analyzing properties of codes with large distance, and show that pseudorandom puncturings still work well in this regime.Comment: Fixed author nam

Convolutional Coded Generalized Direct Sequence Spread Spectrum

In this thesis we investigate the worst-case performance of coded ordinary and coded generalized direct sequence spread spectrum (DSSS) systems in a communication channel corrupted by an unknown and arbitrary interfering signal of bounded power. We consider convolutional codes with Viterbi decoding in order to compare the performance of coded ordinary and coded generalized DSSS systems. For the generalized DSSS system, we use a pulse stream of +1,-1 and 0 as the spreading sequence, which is different from ordinary DSSS system which uses the typical sequence with pulse values of +1 and -1. A C program for performing Monte-Carlo simulations is written in order to evaluate and compare the performance of coded ordinary and coded generalized DSSS systems. Plots of the worst-case error probability versus signal-to-interference ratio are presented for different code rates and constraint lengths of the convolutional code. Simulation results of the worst-case performance of ordinary and generalized DSSS show that generalized DSSS consistently performs appreciably better than ordinary DSSS. Simulation is performed for various code rates, various constraint lengths of the convolutional code and various lengths of the convolutional interleaver. Over all these simulations, it is observed that the difference between ordinary and generalized DSSS gets more pronounced as the channel gets wors