LDPC Codes Based on Latin Squares: Cycle Structure, Stopping Set, and Trapping Set Analysis

It is well known that certain combinatorial structures in the Tanner graph of a low-density parity-check (LDPC) code exhibit a strong influence on its performance under iterative decoding. These structures include cycles, stopping/trapping sets, and parameters such as the diameter of the code. In general, it is very hard to find a complete characterization of such configurations in an arbitrary code, and even harder to understand the intricate relationships that exist between these entities. It is, therefore, of interest to identify a simple setting in which all the described combinatorial structures can be enumerated and studied within a joint framework. One such setting is developed in this paper, for the purpose of analyzing the distribution of short cycles and the structure of stopping and trapping sets in Tanner graphs of LDPC codes based on idempotent and symmetric Latin squares. The parity-check matrices of LDPC codes based on Latin squares have a special form that allows for connecting combinatorial parameters of the codes with the number of certain subrectangles in the Latin squares. Subrectangles of interest can be easily identified, and in certain instances, completely enumerated. This study can be extended in several different directions, one of which is concerned with modifying the code design process in order to eliminate or reduce the number of configurations bearing a negative influence on the performance of the code. Another application of the results includes determining to which extent a configuration governs the behavior of the bit-error rate curve in the waterfall and error-floor regions     

Promising Perspectives on the Use of Fullerenes as Efficient Containers for Beryllium Atoms

The possibility of using fullerenes as containers for toxic beryllium atoms is studied by a multi-scale approach in which first-principles and classical molecular dynamics simulations are combined. By studying the energetics, electronic and spectroscopic properties of Be-fullerene systems and by simulating their interaction at finite temperature in vacuo and in representative biological environments it is concluded that: i) Be endohedral complexes can be obtained by implanting Be atoms at energies >2.3 eV that is consistent with laser implantation technologies; ii) it is in principle possible to distinguish stable endohedral complexes from metastable exohedral ones by optical absorption, suggesting that optical spectroscopy can be a valuable a non-destructive technique to assist the synthesis and the control of implanted films iii) the Be-endohedral complexes are long-lived and thermodynamically stable and can confine beryllium both in vacuo and in aqueous solution; iv) Be@C60 complexes are likely unable to penetrate the selectivity filters of a prototypical protein showing that fullerene prevents undesired interactions with biomolecules and toxicity effects of Be²⁺ related to replacement of the Ca²⁺. Overall, these results provide an assessment on the possibility to encapsulate Be atoms into fullerenes by ion implantation to synthesize inert and highly stable and safe molecular containers for toxic beryllium radionuclides. Great opportunities are expected for the realization and application of Be-C60 complexes to nanotechnology and nanomedicine with particularly appealing perspectives in the field of neutron capture therapy of cancer.