Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

Covering of Subspaces by Subspaces

Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph \cG_q(n,k), $k \geq r$, are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, $q$-analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for $q=2$ with $r=2$ or $r=3$. We discuss the density for some of these coverings. Tables for the best known coverings, for $q=2$ and $5 \leq n \leq 10$, are presented. We present some questions concerning possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352

Measurements of Hadronic Asymmetries in e+e- Collisions

High precision experimental new electroweak results measured at the four LEP experiments and the SLD collaboration are discussed. Heavy quark (b b-bar and c c-bar) forward-backward asymmetries measured at LEP are presented along with polarized forward-backward and left-right asymmetries measured at SLD. The results are compared, and the combined averages are used to evaluate the Standard Model parameters.Comment: 8 Pages, 5 eps files, Presented at the International Europhysics Conference on High Energy Physics, August 1997, Jerusalem, Israe

Experimental Studies of Electroweak Physics

Some experimental new Electroweak physics results measured at the LEP/SLD and the TEVATRON are discussed. The excellent accuracy achieved by the experiments still yield no significant evidence for deviation from the Standard Model predictions, or signal to physics beyond the Standard Model. The Higgs particle still has not been discovered and a low bound is given to its mass.Comment: 10 pages, 5 figures, Invited talk at "Fundemental Particles and Interactions", Vanderbilt University, Nashville, Tennessee, May 199

Folding, Tiling, and Multidimensional Coding

Folding a sequence $S$ into a multidimensional box is a method that is used to construct multidimensional codes. The well known operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes. The new definition of folding is based on lattice tiling and a direction in the $D$-dimensional grid. There are potentially $\frac{3^D-1}{2}$ different folding operations. Necessary and sufficient conditions that a lattice combined with a direction define a folding are given. The immediate and most impressive application is some new lower bounds on the number of dots in two-dimensional synchronization patterns. This can be also generalized for multidimensional synchronization patterns. We show how folding can be used to construct multidimensional error-correcting codes and to generate multidimensional pseudo-random arrays