Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.     

Some new results on dimension and Bose distance for various classes of BCH codes

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over ${\mathbb{F}}_q$ with designed distance $\delta =a{q}^{m-1}-1(\text{resp. }\delta =a\frac{q^m-1}{q-1})$ for all $1\le a\le q-1$, where q is a prime power and $m>1$ is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16], and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range $[a\frac{q^m-1}{q-1},a\frac{q^m-1}{q-1}+T]$ for $0\le a\le q-2$, where $T=q^{\frac{m+1}{2}}-1$ if m is odd, and $T=2q^{\frac{m}{2}}-1$ if m is even.     

Large blocking sets in PG(2,q2)

Minimal blocking sets in $\mathrm{PG}(2,q^2)$ have size at most $q^3+1$. This result is due to Bruen and Thas and the bound is sharp, sets attaining this bound are called unitals. In this paper, we show that the second largest minimal blocking sets have size at most $q^3+1-(p-3)/2$, if $q=p$, $p\ge 67$, or $q=p^h$, $p>7$, $h>1$. Our proof also works for sets having at least one tangent at each of its points (that is, for tangency sets).     

Intersections between the norm-trace curve and some low degree curves

In this paper we analyze the intersection between the norm-trace curve over ${\mathbb{F}}_{q^3}$ and the curves of the form $y=a{x}^3+b{x}^2+cx+d$, giving a complete characterization of the intersection between the curve and the parabolas (a=0), as well as sharp bounds for the other cases. This information is used for the determination of the weight distribution of some one-point AG codes arising from the curve.     

On a conjecture about a class of permutation quadrinomials

Very recently, Tu et al. presented a sufficient condition on $(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x)=x^{3\cdot 2^m}+a_1{x}^{2^{m+1}+1}+a_2{x}^{2^m+2}+a_3{x}^3$ is a class of permutation polynomials over ${\mathbb{F}}_{2^n}$ with $n=2m$ and m odd. In this present paper, we prove that the sufficient condition is also necessary.     

Two solutions for a singular elliptic equation with critical growth at infinity

We look for positive solutions for the singular equation $-\Delta u-\frac{1}{2}\left(x\cdot \nabla u\right)=\mu h\left(x\right)u^{q-1}+\lambda u+u^{(N+2)/(N-2)},\text{in }{\mathbb{R}}^N,$ where $N\ge 3$, $\lambda >0$, $\mu >0$ is a parameter, $0<q<1$ and $h$ has some summability properties. By using a perturbation method and critical point theory, we obtain two solutions when $max\{1,N/4\}<\lambda <N/2$ and the parameter $\mu >0$ is small.     

An upper bound on the number of frequency hypercubes

A frequency n-cube $F^n(q;l_0,...,l_{m-1})$ is an n-dimensional q-by-...-by-q array, where $q=l_0+...+l_{m-1}$, filled by numbers $0,...,m-1$ with the property that each line contains exactly $l_i$ cells with symbol i, $i=0,...,m-1$ (a line consists of q cells of the array differing in one coordinate). The trivial upper bound on the number of frequency n-cubes is $m^{{(q-1)}^n}$. We improve that lower bound for $n>2$, replacing $q-1$ by a smaller value s, by constructing a testing set of size $s^n$ for frequency n-cubes (a testing set is a collection of cells of an array the values in which uniquely determine the array with given parameters). We also construct new testing sets for generalized frequency n-cubes, which are essentially correlation-immune functions in n q-valued arguments; the cardinalities of new testing sets are smaller than for testing sets known before.     

Waring-Hilbert problem on Cantor sets

Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set $C$. It is shown that, for each $m\ge 3$, every real number in the unit interval $[0,1]$ is the sum $x_1^m+x_2^m+?+x_n^m$ with each $x_j$ in $C$ and some $n\le 6^m$. Furthermore, every real number $x$ in the interval $[0,8]$ can be written as $x=x_1^3+x_2^3+?+x_8^3$, the sum of eight cubic powers with each $x_j$ in $C$. Another Cantor set $C\times C$ is also considered. More specifically, when $C\times C$ is embedded into the complex plane $?$, the Waring–Hilbert problem on $C\times C$ has a positive answer for powers less than or equal to 4.