An infinite family of m-ovoids of Q(4,q)

In this paper, we construct an infinite family of $\frac{q-1}{2}$-ovoids of the generalized quadrangle $Q(4,q)$, for $q\equiv 1(\text{mod}4)$ and $q>5$. Together with [3] and [11], this establishes the existence of $\frac{q-1}{2}$-ovoids in $Q(4,q)$ for each odd prime power q.     

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.     

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.     

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.     

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 $n=1$.     

Asymptotic expansions of Gurland's ratio and sharp bounds for their remainders

In this paper, we establish a new asymptotic expansion of Gurland's ratio of gamma functions, that is, as $x\to \infty$,$\frac{\mathrm{\Gamma }(x+p)\mathrm{\Gamma }(x+q)}{\mathrm{\Gamma }{(x+(p+q)/2)}^2}=\exp ?[\sum _{k=1}^n\frac{B_{2k}\left(s\right)?B_{2k}(1/2)}{k(2k?1){(x+r_0)}^{2k?1}}+R_n(x;p,q)]$where $p,q\in \mathbb{R}$ with $w=|p?q|\ne 0$ and $s=(1?w)/2$, $r_0=(p+q?1)/2$, $B_{2n+1}\left(s\right)$ are the Bernoulli polynomials. Using a double inequality for hyperbolic functions, we prove that the function $x?{(?1)}^n{R}_n(x;p,q)$ is completely monotonic on $(?r_0,\infty )$ if $|p?q|<1$, which yields a sharp upper bound for $\left|R_n(x;p,q)\right|$. This shows that the approximation for Gurland's ratio by the truncation of the above asymptotic expansion has a very high accuracy. We also present sharp lower and upper bounds for Gurland's ratio in terms of the partial sum of hypergeometric series. Moreover, some known results are contained in our results when $q\to p$.