The Adjacency Graphs of Some Feedback Shift Registers

The adjacency graphs of feedback shift registers (FSRs) with characteristic function of the form g=(x_0+x_1)*f are considered in this paper. Some properties about these FSRs are given. It is proved that these FSRs contains only prime cycles and these cycles can be divided into two sets such that each set contains no adjacent cycles. When f is a linear function, more properties about these FSRs are derived. It is shown that, when f is a linear function and contains an odd number of terms, the adjacency graph of \mathrm{FSR}((x_0+x_1)*f) can be determined directly from the adjacency graph of \mathrm{FSR}(f). As an application of these results, we determine the adjacency graphs of \mathrm{FSR}((1+x)^4p(x)) and \mathrm{FSR}((1+x)^5p(x)), where p(x) is a primitive polynomial, and construct a large class of de Bruijn sequences from them

De Bruijn Sequences, Adjacency Graphs and Cyclotomy

We study the problem of constructing De Bruijn sequences by joining cycles of linear feedback shift registers (LFSRs) with reducible characteristic polynomials. The main difficulty for joining cycles is to find the location of conjugate pairs between cycles, and the distribution of conjugate pairs in cycles is defined to be adjacency graphs. Let l(x) be a characteristic polynomial, and l(x)=l_1(x)l_2(x)\cdots l_r(x) be a decomposition of l(x) into pairwise co-prime factors. Firstly, we show a connection between the adjacency graph of FSR(l(x)) and the association graphs of FSR(l_i(x)), 1\leq i\leq r. By this connection, the problem of determining the adjacency graph of FSR(l(x)) is decomposed to the problem of determining the association graphs of FSR(l_i(x)), 1\leq i\leq r, which is much easier to handle. Then, we study the association graphs of LFSRs with irreducible characteristic polynomials and give a relationship between these association graphs and the cyclotomic numbers over finite fields. At last, as an application of these results, we explicitly determine the adjacency graphs of some LFSRs, and show that our results cover the previous ones

On Binary de Bruijn Sequences from LFSRs with Arbitrary Characteristic Polynomials

We propose a construction of de Bruijn sequences by the cycle joining method from linear feedback shift registers (LFSRs) with arbitrary characteristic polynomial $f(x)$. We study in detail the cycle structure of the set $\Omega(f(x))$ that contains all sequences produced by a specific LFSR on distinct inputs and provide a fast way to find a state of each cycle. This leads to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach is practical to generate a large class of de Bruijn sequences up to order $n \approx 20$. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction

A number theoretic view on binary shift registers

We describe a number theoretic view on binary shift registers. We illustrate this approach on some basic shift registers by revisiting known and obtaining new results, which we prove using tools from basic number theory, including modular arithmetic.publishedVersio

Mixing multi-core CPUs and GPUs for scientific simulation software

Recent technological and economic developments have led to widespread availability of multi-core CPUs and specialist accelerator processors such as graphical processing units (GPUs). The accelerated computational performance possible from these devices can be very high for some applications paradigms. Software languages and systems such as NVIDIA's CUDA and Khronos consortium's open compute language (OpenCL) support a number of individual parallel application programming paradigms. To scale up the performance of some complex systems simulations, a hybrid of multi-core CPUs for coarse-grained parallelism and very many core GPUs for data parallelism is necessary. We describe our use of hybrid applica- tions using threading approaches and multi-core CPUs to control independent GPU devices. We present speed-up data and discuss multi-threading software issues for the applications level programmer and o er some suggested areas for language development and integration between coarse-grained and ne-grained multi-thread systems. We discuss results from three common simulation algorithmic areas including: partial di erential equations; graph cluster metric calculations and random number generation. We report on programming experiences and selected performance for these algorithms on: single and multiple GPUs; multi-core CPUs; a CellBE; and using OpenCL. We discuss programmer usability issues and the outlook and trends in multi-core programming for scienti c applications developers