Abstract:
Given a nondegenerate ternary form $f=f(x_1,x_2,x_3)$ of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces to construct a certain positive-dimensional family of irreducible representations of the generalized Clifford algebra associated to $f.$ From this we obtain the existence of linear Pfaffian representations of the quartic surface $X_f=\{w^4=f(x_1,x_2,x_3)\},$ as well as information on the Brill-Noether theory of a general smooth curve in the linear system $|\mathcal{O}_{X_f}(3)|.$

Abstract:
Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters $[3^m-1, 3^m-1-2m, 4]$ by Carlet, Ding and Yuan in 2005. In this paper, almost perfect nonlinear monomials, and a number of other monomials over $\gf(3^m)$ are used to construct optimal ternary cyclic codes with the same parameters. Nine open problems on such codes are also presented.

Abstract:
Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Let $m=2\ell+1$ for an integer $\ell\geq 1$ and $\pi$ be a generator of $\gf(3^m)^*$. In this paper, a class of cyclic codes $\C_{(u,v)}$ over $\gf(3)$ with two nonzeros $\pi^{u}$ and $\pi^{v}$ is studied, where $u=(3^m+1)/2$, and $v=2\cdot 3^{\ell}+1$ is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over $\gf(3^m)$, the cyclic code $\C_{(u,v)}$ is shown to have minimal distance four, which is the best minimal distance for any linear code over $\gf(3)$ with length $3^m-1$ and dimension $3^m-1-2m$ according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied.

Abstract:
This work is twofold. First, the largest minimum distance of a ternary cyclic codes of parameters ${[n, \frac{n}{2}]}$ , is determined for n even, not a multiple of 3, by using the Chen algorithm, for n = 26, 34, 38, 46, 50, 58, 62, 68, 70, 74. Next, seven new classes of isodual ternary cyclic codes are introduced for n singly even, not a multiple of 3.

Abstract:
As a subclass of linear codes, cyclic codes have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, five families of three-weight ternary cyclic codes whose duals have two zeros are presented. The weight distributions of the five families of cyclic codes are settled. The duals of two families of the cyclic codes are optimal.

Abstract:
Cyclic codes are an important subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics. In this paper, two families of optimal ternary cyclic codes are presented. The first family of cyclic codes has parameters $[3^m-1, 3^m-1-2m, 4]$ and contains a class of conjectured cyclic codes and several new classes of optimal cyclic codes. The second family of cyclic codes has parameters $[3^m-1, 3^m-2-2m, 5]$ and contains a number of classes of cyclic codes that are obtained from perfect nonlinear functions over $\fthreem$, where $m>1$ and is a positive integer.

Abstract:
This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex hull of a plane quartic to be exactly semidefinite representable with at most 12 lifting variables. If the quartic is rationally parametrizable, an exact semidefinite representation with 2 lifting variables can be obtained. Various numerical examples illustrate the techniques and suggest further research directions.

Abstract:
Using fixed point methods, we prove the stability and superstability of -ternary additive, quadratic, cubic, and quartic homomorphisms in -ternary rings for the functional equation , for each .

Abstract:
A correspondence between quartic étale algebras over a field and quadratic étale extensions of cubic étale algebras is set up and investigated. The basic constructions are laid out in general for sets with a profinite group action and for torsors, and translated in terms of étale algebras and Galois algebras. A parametrization of cyclic quartic algebras is given.

Abstract:
The dynamic complexity of the reachability query is studied in the dynamic complexity framework of Patnaik and Immerman, restricted to quantifier-free update formulas. It is shown that, with this restriction, the reachability query cannot be dynamically maintained, neither with binary auxiliary relations nor with unary auxiliary functions, and that ternary auxiliary relations are more powerful with respect to graph queries than binary auxiliary relations. Further inexpressibility results are given for the reachability query in a different setting as well as for a syntactical restriction of quantifier-free update formulas. Moreover inexpressibility results for some other queries are presented.