Implementing Line-Hermitian Grassmann codes

In [I. Cardinali and L. Giuzzi. Line Hermitian Grassmann codes and their parameters. Finite Fields Appl., 51: 407-432, 2018] we introduced line Hermitian Grassmann codes and determined their parameters. The aim of this paper is to present (in the spirit of [I. Cardinali and L. Giuzzi. Enumerative coding for line polar Grassmannians with applications to codes. Finite Fields Appl., 46:107-138, 2017]) an algorithm for the point enumerator of a line Hermitian Grassmannian which can be usefully applied to get efficient encoders, decoders and error correction algorithms for the aforementioned codes.Comment: 26 page

Direct constructions of hyperplanes of dual polar spaces arising from embeddings

Let e be one of the following full projective embeddings of a finite dual polar space Delta of rank n >= 2: (i) The Grassmann-embedding of the symplectic dual polar space Delta congruent to DW(2n 1,q); (ii) the Grassmann-embedding of the Hermitian dual polar space Delta congruent to DH(2n-1, q(2)); (iii) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space Delta congruent to DQ(-)(2n+ 1, q). Let H-e denote the set of all hyperplanes of Delta arising from the embedding e. We give a method for constructing the hyperplanes of H-e without implementing the embedding e and discuss (possible) applications of the given construction

Line Hermitian Grassmann codes and their parameters

In this paper we introduce and study line Hermitian Grassmann codes as those subcodes of the Grassmann codes associated to the 2-Grassmannian of a Hermitian polar space defined over a finite field. In particular, we determine the parameters and characterize the words of minimum weight

Construction of equiangular signatures for synchronous CDMA systems

Welch bound equality (WBE) signature sequences maximize the uplink sum capacity in direct-spread synchronous code division multiple access (CDMA) systems. WBE sequences have a nice interference invariance property that typically holds only when the system is fully loaded, and, to maintain this property, the signature set must be redesigned and reassigned as the number of active users changes. An additional equiangular constraint on the signature set, however, maintains interference invariance. Finding such signatures requires equiangular side constraints to be imposed on an inverse eigenvalue problem. The paper presents an alternating projection algorithm that can design WBE sequences that satisfy equiangular side constraints. The proposed algorithm can be used to find Grassmannian frames as well as equiangular tight frames. Though one projection is onto a closed, but non-convex, set, it is shown that this algorithm converges to a fixed point, and these fixed points are partially characterized

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

A superspace module for the FeynRules package

We describe an additional module for the Mathematica package FeynRules that allows for an easy building of any N=1 supersymmetric quantum field theory, directly in superspace. After the superfield content of a specific model has been implemented, the user can study the properties of the model, such as the supersymmetric transformation laws of the associated Lagrangian, directly in Mathematica. While the model dependent parts of the latter, i.e., the soft supersymmetry-breaking Lagrangian and the superpotential, have to be provided by the user, the model independent pieces, such as the gauge interaction terms, are derived automatically. Using the strengths of the Feynrules program, it is then possible to derive all the Feynman rules associated to the model and implement them in all the Feynman diagram calculators interfaced to FeynRules in a straightforward way.Comment: 54 pages, 9 tables, version accepted by CP

Algorithms in Lattice QCD

The enormous computing resources that large-scale simulations in Lattice QCD require will continue to test the limits of even the largest supercomputers into the foreseeable future. The efficiency of such simulations will therefore concern practitioners of lattice QCD for some time to come. I begin with an introduction to those aspects of lattice QCD essential to the remainder of the thesis, and follow with a description of the Wilson fermion matrix M, an object which is central to my theme. The principal bottleneck in Lattice QCD simulations is the solution of linear systems involving M, and this topic is treated in depth. I compare some of the more popular iterative methods, including Minimal Residual, Corij ugate Gradient on the Normal Equation, BI-Conjugate Gradient, QMR., BiCGSTAB and BiCGSTAB2, and then turn to a study of block algorithms, a special class of iterative solvers for systems with multiple right-hand sides. Included in this study are two block algorithms which had not previously been applied to lattice QCD. The next chapters are concerned with a generalised Hybrid Monte Carlo algorithm (OHM C) for QCD simulations involving dynamical quarks. I focus squarely on the efficient and robust implementation of GHMC, and describe some tricks to improve its performance. A limited set of results from HMC simulations at various parameter values is presented. A treatment of the non-hermitian Lanczos method and its application to the eigenvalue problem for M rounds off the theme of large-scale matrix computations