A new explicit time-reversible orbit integrator for the equations of motion in a static homogeneous magnetic field —called Cyclotronic integrator— is presented. Like Spreiter and Walter’s Taylor expansion algorithm, for sufficiently weak electric field gradients this second order method does not require a fine resolution of the Larmor motion; it has however the essential advantage of being symp...

In this paper we propose a heuristic technique for distributing points on the surface of a unit n-dimensional Euclidean sphere, generated as the orbit of a finite cyclic subgroup of orthogonal matrices, the so called cyclic group codes. Massive numerical experiments were done and many new cyclic group codes have been obtained in several dimensions at various rate. The obtained results assure th...

Orbits of graphs under local complementation (LC) and edge local complementation (ELC) have been studied in several different contexts. For instance, there are connections between orbits of graphs and errorcorrecting codes. We define a new graph class, ELC-preserved graphs, comprising all graphs that have an ELC orbit of size one. Through an exhaustive search, we find all ELC-preserved graphs o...

Orbits of graphs under the operation edge local complementation (ELC) are defined. We show that the ELC orbit of a bipartite graph corresponds to the equivalence class of a binary linear code. The information sets and the minimum distance of a code can be derived from the corresponding ELC orbit. By extending earlier results on local complementation (LC) orbits, we classify the ELC orbits of al...

We present a symmetric LDPC code with constant rate and constant distance (i.e. good LDPC code) that its constraint space is generated by the orbit of one constant weight constraint under a group action. Our construction provides the first symmetric LDPC good codes. This solves the main open problem raised by Kaufman and Wigderson in [4].

We consider the billiard map in a unit cube of R, and we compute the complexity of a word which codes an orbit in direction ω. Under some hypothesis on the direction, we obtain an exact formula and shows that the order of magnitude is n.

The article explores the syndrome invariants of АГ-group automorphisms Reed–Solomon codes (RS-codes) that are a joint group affine and cyclic permutations. found real set norms N Г-orbits make up one or another АГ-orbit. vectors with 2 1 Cδ− coordinates from Galois field, determined by all kinds pairs components error syndromes. In this form, АГ-orbits were cumbersome difficult to use. Therefor...

in this paper, we consider a class of column-weight two quasi-cyclic low-density paritycheck codes in which the girth can be large enough, as an arbitrary multiple of 8. then we devote a convolutional form to these codes, such that their generator matrix can be obtained by elementary row and column operations on the parity-check matrix. finally, we show that the free distance of the convolution...

This paper describes tracking studies of FFAGs and radial-sector cyclotrons with reverse bends using the cyclotron equilibrium orbit code CYCLOPS. The results for FFAGs confirm those obtained with lumped-element codes, and suggest that cyclotron codes will prove to be important tools for evaluating the measured fields of FFAG magnets. The results for radial-sector cyclotrons show that the use o...