Sub-quadratic Decoding of One-point Hermitian Codes

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realisation of the Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimisation. The second is a Power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the same methods from computer algebra, yielding similar asymptotic complexities.Comment: New version includes simulation results, improves some complexity results, as well as a number of reviewer corrections. 20 page

Affine Grassmann Codes

We consider a new class of linear codes, called affine Grassmann codes. These can be viewed as a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. We determine the length, dimension, and the minimum distance of any affine Grassmann code. Moreover, we show that affine Grassmann codes have a large automorphism group and determine the number of minimum weight codewords.Comment: Slightly Revised Version; 18 page

Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields

About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the maximum number of common zeros that $r$ linearly independent homogeneous polynomials of degree $d$ in $m+1$ variables with coefficients in a finite field with $q$ elements can have in the corresponding $m$-dimensional projective space. Recently, it has been shown by Datta and Ghorpade that this conjecture is valid if $r$ is at most $m+1$ and can be invalid otherwise. Moreover a new conjecture was proposed for many values of $r$ beyond $m+1$. In this paper, we prove that this new conjecture holds true for several values of $r$. In particular, this settles the new conjecture completely when $d=3$. Our result also includes the positive result of Datta and Ghorpade as a special case. Further, we determine the maximum number of zeros in certain cases not covered by the earlier conjectures and results, namely, the case of $d=q-1$ and of $d=q$. All these results are directly applicable to the determination of the maximum number of points on sections of Veronese varieties by linear subvarieties of a fixed dimension, and also the determination of generalized Hamming weights of projective Reed-Muller codes.Comment: 15 page

Linear Codes associated to Determinantal Varieties

We consider a class of linear codes associated to projective algebraic varieties defined by the vanishing of minors of a fixed size of a generic matrix. It is seen that the resulting code has only a small number of distinct weights. The case of varieties defined by the vanishing of 2 x 2 minors is considered in some detail. Here we obtain the complete weight distribution. Moreover, several generalized Hamming weights are determined explicitly and it is shown that the first few of them coincide with the distinct nonzero weights. One of the tools used is to determine the maximum possible number of matrices of rank 1 in a linear space of matrices of a given dimension over a finite field. In particular, we determine the structure and the maximum possible dimension of linear spaces of matrices in which every nonzero matrix has rank 1.Comment: 12 pages; to appear in Discrete Mat

Routing for analog chip designs at NXP Semiconductors

During the study week 2011 we worked on the question of how to automate certain aspects of the design of analog chips. Here we focused on the task of connecting different blocks with electrical wiring, which is particularly tedious to do by hand. For digital chips there is a wealth of research available for this, as in this situation the amount of blocks makes it hopeless to do the design by hand. Hence, we set our task to finding solutions that are based on the previous research, as well as being tailored to the specific setting given by NXP. This resulted in an heuristic approach, which we presented at the end of the week in the form of a protoype tool. In this report we give a detailed account of the ideas we used, and describe possibilities to extend the approach

Bounding the number of points on a curve using a generalization of Weierstrass semigroups

In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the (generalized) Weierstrass semigroup [P. Beelen, N. Tuta\c{s}: A generalization of the Weierstrass semigroup, J. Pure Appl. Algebra, 207(2), 2006] for an $n$-tuple of places is known, even if the exact defining equation of the curve is not known. As shown in examples, this sometimes enables one to get an upper bound for the number of rational places for families of function fields. Our results extend results in [O. Geil, R. Matsumoto: Bounding the number of $\mathbb{F}_q$-rational places in algebraic function fields using Weierstrass semigroups. Pure Appl. Algebra, 213(6), 2009]

The dust emission of high-redshift quasars

The detection of powerful near-infrared emission in high redshift (z>5) quasars demonstrates that very hot dust is present close to the active nucleus also in the very early universe. A number of high-redshift objects even show significant excess emission in the rest frame NIR over more local AGN spectral energy distribution (SED) templates. In order to test if this is a result of the very high luminosities and redshifts, we construct mean SEDs from the latest SDSS quasar catalogue in combination with MIR data from the WISE preliminary data release for several redshift and luminosity bins. Comparing these mean SEDs with a large sample of z>5 quasars we could not identify any significant trends of the NIR spectral slope with luminosity or redshift in the regime 2.5 < z < 6 and 10^45 < nuL_nu(1350AA) < 10^47 erg/s. In addition to the NIR regime, our combined Herschel and Spitzer photometry provides full infrared SED coverage of the same sample of z>5 quasars. These observations reveal strong FIR emission (L_FIR > 10^13 L_sun) in seven objects, possibly indicating star-formation rates of several thousand solar masses per year. The FIR excess emission has unusally high temperatures (T ~ 65 K) which is in contrast to the temperature typically expected from studies at lower redshift (T ~ 45 K). These objects are currently being investigated in more detail.Comment: 6 pages, 3 figures, to appear in the proceedings to "The Central Kiloparsec in Galactic Nuclei (AHAR2011)", Journal of Physics: Conference Series (JPCS), IOP Publishin

Automorphism groups of Grassmann codes

We use a theorem of Chow (1949) on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question of Beelen et al. (2010) concerning the permutation automorphism groups of affine Grassmann codes. Finally, we prove an analogue of Chow's theorem for the case of Schubert divisors in Grassmannians and then use it to determine the automorphism group of linear codes associated to such Schubert divisors. In the course of this work, we also give an alternative short proof of MacWilliams theorem concerning the equivalence of linear codes and a characterization of maximal linear subspaces of Schubert divisors in Grassmannians.Comment: revised versio

Transaction Costs for Design-Build-Finance-Maintain Contracts

This paper gives insight in how transaction costs arise and how in theory transaction costs can be reduced. A comparison between theory and practice has been made. A study of a case in the Netherlands, the Second Coentunnel showed how transaction costs in practice appear, in which stage of the purchasing process these cost arise and also how transaction costs can be reduced. Cost specifications, handed by the public and private parties, make clear that in every phase of the process the client makes expenses. The client spends the most money during the initiative phase. The private parties start making costs in the first phase of the tender (prequalification). For contractors the most expensive phase is the dialogue phase. Taking all the costs in overview, noticeable is that all of the costs are related to the duration of the different phases of the process and required capacity of personnel. Success factors from theory and practice have been identified in the process in which transaction costs arise. Theory and practice have been compared and resulted in a list of twelve success factors. By implementing these success factors in future projects the expectation is that transaction costs will not be unnecessary high

Duals of Affine Grassmann Codes and their Relatives

Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work [2]. Here we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results of [2] concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes.Comment: 20 page