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.     

Classification of some quadrinomials over finite fields of odd characteristic

In this paper, we completely determine all necessary and sufficient conditions such that the polynomial $f(x)=x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $a,b,c\in {\mathbb{F}}_q^{⁎}$, is a permutation quadrinomial of ${\mathbb{F}}_{q^2}$ over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where $char({\mathbb{F}}_q)=2$ and finally, in [16], Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial $x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $char({\mathbb{F}}_q)=3,5$ and $a,b,c\in {\mathbb{F}}_q^{⁎}$ and proposed some new classes of permutation quadrinomials of ${\mathbb{F}}_{q^2}$.In particular, in this paper we classify all permutation polynomials of ${\mathbb{F}}_{q^2}$ of the form $f(x)=x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $a,b,c\in {\mathbb{F}}_q^{⁎}$, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials.     

A note on complex conference matrices

Let ${\mathbb{F}}_q$ be the Galois field of order $q=p^m$, p a prime number and m a positive integer. We prove in this article that for any nontrivial multiplicative character ϰ of ${\mathbb{F}}_q^{⁎}$ and for any $b\in {\mathbb{F}}_q^{⁎}$ we have$\sum _{a\in {\mathbb{F}}_q^{⁎}}\varkappa (a)\overline{\varkappa (a+b)}=-1$. Whenever q is odd and ϰ is the Legendre symbol this formula reduces to the well-known Jacobsthal's formula. A complex conference matrix is a square matrix of order n with zero diagonal and unimodular complex numbers elsewhere such that $C^{⁎}C=(n-1)I$. Paley used finite fields with odd orders $q=p^m$, p prime and the real Legendre symbol to construct real symmetric conference matrices of orders $q+1$ whenever $q\equiv 1(\mathrm{mod}4)$ and real skew-symmetric conference matrices of orders $q+1$ whenever $q\equiv -1(\mathrm{mod}4)$. In this article we extend Paley construction to the complex setting. We extend Jacobsthal's formula to all other nontrivial characters to produce a complex symmetric conference matrix of order $q+1$ whenever $q\ge 4$ is any prime power as well as a complex skew-symmetric conference matrix of order $q+1$ whenever q is any odd prime power. These matrices were constructed very recently in connection with harmonic Grassmannian codes, by use of finite fields and the character table of their additive characters. We propose here a new proof of their construction by use of the above generalized formula similarly as was done by Paley in the real case. We also classify, up to equivalence, the complex conference matrices constructed with some nontrivial characters. In particular, we prove that the complex conference matrix constructed with any nontrivial multiplicative character ϰ and that one constructed with ${\varkappa }^{p^k}$ for any integer $k=1,...m-1$ are permutation equivalent. Moreover, we determine the spectrum of any complex conference matrix obtained from this construction.     

New results on permutation binomials of finite fields

After a brief review of the existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then focus on PBs of ${\mathbb{F}}_{q^2}$ of the form $X^n(X^{d(q-1)}+a)$, where n and d are positive integers and $a\in {\mathbb{F}}_{q^2}^{⁎}$. Our contributions include two nonexistence results: (1) If q is even and sufficiently large and $a^{q+1}\ne 1$, then $X^n(X^{3(q-1)}+a)$ is not a PB of ${\mathbb{F}}_{q^2}$. (2) If $2\le d|q+1$, q is sufficiently large and $a^{q+1}\ne 1$, then $X^n(X^{d(q-1)}+a)$ is not a PB of ${\mathbb{F}}_{q^2}$ under certain additional conditions. (1) partially confirms a recent conjecture by Tu et al. (2) is an extension of a previous result with $n=1$.