Generalized Hermitian Codes over GF(2^r)

In this paper we studied generalization of Hermitian function field proposed by A.Garcia and H.Stichtenoth. We calculated a Weierstrass semigroup of the point at infinity for the case q=2, r>=3. It turned out that unlike Hermitian case, we have already three generators for the semigroup. We then applied this result to codes, constructed on generalized Hermitian function fields. Further, we applied results of C.Kirfel and R.Pellikaan to estimating a Feng-Rao designed distance for GH-codes, which improved on Goppa designed distance. Next, we studied the question of codes dual to GH-codes. We identified that the duals are also GH-codes and gave an explicit formula. We concluded with some computational results. In particular, a new record-giving [32,16,>=12]-code over GF(8) was presented

Towards a Better Understanding of the Semigroup Tree

In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which of the two types takes place in particular for well-known classes of semigroups. Also we study the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof. We conclude with some thoughts that show how this study of the semigroup tree may help in solving the conjecture of Fibonacci-like behavior of the number of semigroups with given genus.Comment: 17 pages, 2 figure

A Note on the Injection Distance

Koetter and Kschischang showed in [R. Koetter and F.R. Kschischang, "Coding for Errors and Erasures in Random Network Coding," IEEE Trans. Inform. Theory, {54(8), 2008] that the network coding counterpart of Gabidulin codes performs asymptotically optimal with respect to the subspace distance. Recently, Silva and Kschischang introduced in [D. Silva and F.R. Kschischang, "On Metrics for Error Correction in Network Coding," To appear in IEEE Trans. Inform. Theory, ArXiv: 0805.3824v4[cs.IT], 2009] the injection distance to give a detailed picture of what happens in noncoherent network coding. We show that the above codes are also asymptotically optimal with respect to this distance

Some Reflection on Gaps and Antinomies

El autor reflexiona en torno al tratamiento que L. FERRAJOLI da a las antinomias y lagunas en Principia juris. Colateralmente aborda a su vez críticamente los comentarios que respecto del mismo aspecto de la obra ferrajoliana han sido formulados por J. J. MORESO. Respecto de ésta, BULYGIN cuestiona, principalmente, i) la distinción entre sistemas de un solo nivel normativo o sistemas nomoestáticos (que dan lugar a lo que llama el Estado legislativo de Derecho) y sistemas nomodinámicos, o de varios niveles normativos (Estado constitucional ); ii) la idea de que las lagunas y antinomias sólo aparecen en los sistemas de más de un nivel normativo, no así en los de un solo nivel; iii) el postulado por el cual de aquello que no está permitida la comisión, está permitida la omisión, y iv) el postulado de que todo comportamiento supone una modalidad que lo califica deónticamente.The author reflects on the subject of L. FERRAJOLI‘s treatment of antinomies and gaps in Principia juris. At the same time, he critically approaches the comments formulated by J. J. MORESO on the same aspect of FERRAJOLI‘s work. With respect to FERRAJOLI‘s work, BULYGIN mainly questions, i) the distinction between systems with a single normative level, or nomostatic systems (which give rise to what he calls Legislative State of law) and systems with various normative levels or nomodynamic systems (Constitutional State); ii) the idea that gaps and antinomies only appear in systems of more than one normative level, and not in those of a single level; iii) the postulate which permits the ommission of actions whose commission is forbidden; and iv) the postulate by which all behaviour presupposes a modality which qualifies it deontically

Toward high-speed effective numerical simulation of multiple filamentation of high-power femtosecond laser radiation in transparent medium

High-power femtosecond laser radiation during the propagation in air (and other transparent media) experiences multiple filamentation. Filamentation is a unique nonlinear optical phenomenon, which is accompanied by a wealth of nonlinear optical effects such as formation of extended plasma channels in the beam wake, generation of higher harmonics and supercontinuum, generation of THz radiation. The manifestations of laser filamentation can be useful for solving atmospheric optics problems related to remote sensing of the environment as well as directed transmission of laser power. The classical numerical methods used for simulating the nonlinear long-range atmospheric propagation of high-power radiation with a sufficiently large laser beam aperture have almost reached their limit regarding the acceleration of calculations. To solve this problem and speed-up the numerical simulations of laser filamentation, we propose an improved numerical technique based on a modified method of phase screens constructed on a sparse spatial grid. Within the framework of this technique, we seek for optimal ansatz (substitution function) to the governing equations using the machine learning technology, which provides for the best correspondence to the numerical solution of the test problem using a denser spatial gri