On quartics with three-divisible sets of cusps

We study the geometry and codes of quartic surfaces with many cusps. We apply Gr\"obner bases to find examples of various configurations of cusps on quartics.Comment: 15 page

Wolf Barth (1942--2016)

In this article we describe the life and work of Wolf Barth who died on 30th December 2016. Wolf Barth's contributions to algebraic variety span a wide range of subjects. His achievements range from what is now called the Barth-Lefschetz theorems to his fundamental contributions to the theory of algebraic surfaces and moduli of vector bundles, and include his later work on algebraic surfaces with many singularities, culminating in the famous Barth sextic.Comment: accepted for publication in Jahresbericht der Deutschen Mathematiker-Vereinigung, obituary, 17 pages, 2 figures, 1 phot

The Nonexistence of [132, 6, 86]3 Codes and [135, 6, 88]3 Codes

We prove the nonexistence of [g3(6, d), 6, d]3 codes for d = 86, 87, 88, where g3(k, d) = ∑⌈d/3i⌉ and i=0 ... k−1. This determines n3(6, d) for d = 86, 87, 88, where nq(k, d) is the minimum length n for which an [n, k, d]q code exists

Waring identifiable subspaces over finite fields

Waring's problem, of expressing an integer as the sum of powers, has a very long history going back to the 17th century, and the problem has been studied in many different contexts. In this paper we introduce the notion of a Waring subspace and a Waring identifiable subspace with respect to a projective algebraic variety $\mathcal X$. When $\mathcal X$ is the Veronese variety, these subspaces play a fundamental role in the theory of symmetric tensors and are related to the Waring decomposition and Waring identifiability of symmetric tensors (homogeneous polynomials). We give several constructions and classification results of Waring identifiable subspaces with respect to the Veronese variety in ${\mathbb{P}}^5({\mathbb{F}}_q)$ and in ${\mathbb{P}}^{9}({\mathbb{F}}_q)$, and include some applications to the theory of linear systems of quadrics in ${\mathbb{P}}^3({\mathbb{F}}_q)$