Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in cite{D} constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions ov
er finite fields and presented ten open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN functions $f(x)=x^{q^m-2}$ and the Dobbertin APN function $f(x)=x^{2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$. From the calculation of linear spans and the minimal polynomials of two sequences generated by these two classes of APN functions, the dimensions of the corresponding cyclic codes are determined and lower bounds on the minimum weight of these cyclic codes are presented. Actually, we present a framework for the minimal polynomial and linear span of the sequence $s^{infty}$ defined by $s_t=Tr((1+alpha^t)^e)$, where $alpha$ is a primitive element in $GF(q)$. These techniques can also be applied into other open problems in cite{D}.
A modified four-state CVQKD protocol is proposed to increase the maximum transmission distance and tolerable excess noise in the presence of Gaussian lossy and noisy channel by using a noiseless linear amplifier (NLA). A NLA with amplitude gain g can
increase the maximum admission losses by 20log(g) dB.
Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case when an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and sufficient con
dition. In this case, we also obtain a divisible property for the weight of a codeword. Further, we present a necessary and sufficient condition for an irreducible cyclic code with only one nonzero weight. Finally, we determine the weight distribution of an irreducible cyclic code for some cases.
Bent functions, which are maximally nonlinear Boolean functions with even numbers of variables and whose Hamming distance to the set of all affine functions equals $2^{n-1}pm 2^{frac{n}{2}-1}$, were introduced by Rothaus in 1976 when he considered pr
oblems in combinatorics. Bent functions have been extensively studied due to their applications in cryptography, such as S-box, block cipher and stream cipher. Further, they have been applied to coding theory, spread spectrum and combinatorial design. Hyper-bent functions, as a special class of bent functions, were introduced by Youssef and Gong in 2001, which have stronger properties and rarer elements. Many research focus on the construction of bent and hyper-bent functions. In this paper, we consider functions defined over $mathbb{F}_{2^n}$ by $f_{a,b}:=mathrm{Tr}_{1}^{n}(ax^{(2^m-1)})+mathrm{Tr}_{1}^{4}(bx^{frac{2^n-1}{5}})$, where $n=2m$, $mequiv 2pmod 4$, $ain mathbb{F}_{2^m}$ and $binmathbb{F}_{16}$. When $ain mathbb{F}_{2^m}$ and $(b+1)(b^4+b+1)=0$, with the help of Kloosterman sums and the factorization of $x^5+x+a^{-1}$, we present a characterization of hyper-bentness of $f_{a,b}$. Further, we use generalized Ramanujan-Nagell equations to characterize hyper-bent functions of $f_{a,b}$ in the case $ainmathbb{F}_{2^{frac{m}{2}}}$.
Cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes, have been studied for many years. However, the weight distributions of cyclic codes are difficult to determine. From elliptic curve
s, this paper determines the weight distributions of dual codes of cyclic codes with two zeros for a few more cases.
