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The pursuit of compact lasers with low-thresholds has imposed strict requirements on tight light confinements with minimized radiation losses. Bound states in the continuum (BICs) have been recently demonstrated as an effective mechanism to trap light along the out-of-plane direction, paving the way to low-threshold lasers. To date, most reported BIC lasers are still bulky due to the absence of in-plane light confinement. In this work, we combine BICs and photonic band gaps to realize three-dimensional (3D) light confinements, as referred to miniaturized (mini-) BICs. Together with 3D carrier confinements provided by quantum dots (QDs) as optical gain materials, we have realized highly-compact active BIC resonators with a record-high quality ($Q$) factor up to 32500, which enables single-mode continuous wave (CW) lasing with the lowest threshold of 80 W/cm$^{2}$ among the reported BIC lasers. In addidtion, our photon statistics measurements under both CW and pulsed excitations confirm the occurence of the phase transition from spontaneous emission to stimulated emission, further suggesting that conventional criteria of input-output and linewidth are not sufficient for claiming nanoscale lasing. Our work reveal a via path towards compact BIC lasers with ultra-low power consumption and potentially boost the applications in cavity quantum electrodynamics (QEDs), nonlinear optics and integrated photonics., Comment: 12 pages,4 figures in maintext

For monolayer transition metal chalcogenides (TMDs), electrons and excitons in different valleys can be driven to opposite directions by the Berry curvature, serving as a valley-dependent effective magnetic field. In addition to monolayer TMDs, Van der Waals heterostructures provide an attractive platform for emerging valley physics and devices with superior physics properties. Interlayer excitons in TMD heterostructures have a long valley lifetime as compared to monolayer intralayer excitons. Here we report an interlayer exciton valley polarization and valley Hall effect in MoS2/WSe2 in room temperature. The separation for excitons with different valley index is observed with polarization-dependent photoluminescence mapping. The exciton separation is perpendicular to their transport directions. The valley Hall effect for indirect excitons is sustained even at room temperature, in contrast with the cryo-temperatures in previous experiments in monolayer TMDs. Room temperature demonstration of the indirect exciton valley polarization and valley Hall effect might open new perspectives for the development of opto-valleytronic devices based on TMD heterostructures.

Two-dimensional transition-metal dichalcogenides (TMDC) are of great interest for on-chip nanophotonics due to their unique optoelectronic properties. Here, we propose and realize coupling of tungsten diselenide (WSe2) monolayers to circular Bragg grating structures to achieve enhanced emission. The interaction between WSe2 and the resonant mode of the structure results in Purcell-enhanced emission, while the symmetric geometrical structure improves the directionality of the out-coupling stream of emitted photons. Furthermore, this hybrid structure produces a record high contrast of the spin valley readout (> 40%) revealed by the polarization resolved photoluminescence (PL) measurements. Our results are promising for on-chip integration of TMDC monolayers with optical resonators for nanophotonic circuits.

In this paper, we further investigate properties of generalized bent Boolean functions from $\Z_{p}^n$ to $\Z_{p^k}$, where $p$ is an odd prime and $k$ is a positive integer. For various kinds of representations, sufficient and necessary conditions for bent-ness of such functions are given in terms of their various kinds of component functions. Furthermore, a subclass of gbent functions corresponding to relative difference sets, which we call $\Z_{p^k}$-bent functions, are studied. It turns out that $\Z_{p^k}$-bent functions correspond to a class of vectorial bent functions, and the property of being $\Z_{p^k}$-bent is much stronger then the standard bent-ness. The dual and the generalized Gray image of gbent function are also discussed. In addition, as a further generalization, we also define and give characterizations of gbent functions from $\Z_{p^l}^n$ to $\Z_{p^k}$ for a positive integer $l$ with $l

In this paper, we investigate properties of functions from $\mathbb{Z}_{p}^n$ to $\mathbb{Z}_q$, where $p$ is an odd prime and $q$ is a positive integer divided by $p$. we present the sufficient and necessary conditions for bent-ness of such generalized Boolean functions in terms of classical $p$-ary bent functions, when $q=p^k$. When $q$ is divided by $p$ but not a power of it, we give an sufficient condition for weakly regular gbent functions. Some related constructions are also obtained.

We propose a general approach to construct cryptographic significant Boolean functions of $(r+1)m$ variables based on the additive decomposition $\mathbb{F}_{2^{rm}}\times\mathbb{F}_{2^m}$ of the finite field $\mathbb{F}_{2^{(r+1)m}}$, where $r$ is odd and $m\geq3$. A class of unbalanced functions are constructed first via this approach, which coincides with a variant of the unbalanced class of generalized Tu-Deng functions in the case $r=1$. This class of functions have high algebraic degree, but their algebraic immunity does not exceeds $m$, which is impossible to be optimal when $r>1$. By modifying these unbalanced functions, we obtain a class of balanced functions which have optimal algebraic degree and high nonlinearity (shown by a lower bound we prove). These functions have optimal algebraic immunity provided a combinatorial conjecture on binary strings which generalizes the Tu-Deng conjecture is true. Computer investigations show that, at least for small values of number of variables, functions from this class also behave well against fast algebraic attacks.

In this paper, we propose a new construction of quadratic bent functions in polynomial forms. Right Euclid algorithm in skew-polynomial rings over finite fields of characteristic 2 is applied in the proof., Comment: This paper has been withdrawn for some errors in page 11

Constructing $2m$-variable Boolean functions with optimal algebraic immunity based on decomposition of additive group of the finite field $\mathbb{F}_{2^{2m}}$ seems to be a promising approach since Tu and Deng's work. In this paper, we consider the same problem in a new way. Based on polar decomposition of the multiplicative group of $\mathbb{F}_{2^{2m}}$, we propose a new construction of Boolean functions with optimal algebraic immunity. By a slight modification of it, we obtain a class of balanced Boolean functions achieving optimal algebraic immunity, which also have optimal algebraic degree and high nonlinearity. Computer investigations imply that this class of functions also behave well against fast algebraic attacks., Comment: 20 pages

A class of bilinear permutation polynomials over a finite field of characteristic 2 was constructed in a recursive manner recently which involved some other constructions as special cases. We determine the compositional inverses of them based on a direct sum decomposition of the finite field. The result generalizes that in [R.S. Coulter, M. Henderson, The compositional inverse of a class of permutation polynomials over a finite field, Bull. Austral. Math. Soc. 65 (2002) 521-526]., Comment: 17 pages

We give new characterizations of the algebra $\mathscr{L}_n(\mathbb{F}_{q^n})$ formed by all linearized polynomials over the finite field $\mathbb{F}_{q^n}$ after briefly surveying some known ones. One isomorphism we construct is between $\mathscr{L}_n(\mathbb{F}_{q^n})$ and the composition algebra $\mathbb{F}_{q^n}^\vee\otimes_{\mathbb{F}_{q}}\mathbb{F}_{q^n}$. The other isomorphism we construct is between $\mathscr{L}_n(\mathbb{F}_{q^n})$ and the so-called Dickson matrix algebra $\mathscr{D}_n(\mathbb{F}_{q^n})$. We also further study the relations between a linearized polynomial and its associated Dickson matrix, generalizing a well-known criterion of Dickson on linearized permutation polynomials. Adjugate polynomial of a linearized polynomial is then introduced, and connections between them are discussed. Both of the new characterizations can bring us more simple approaches to establish a special form of representations of linearized polynomials proposed recently by several authors. Structure of the subalgebra $\mathscr{L}_n(\mathbb{F}_{q^m})$ which are formed by all linearized polynomials over a subfield $\mathbb{F}_{q^m}$ of $\mathbb{F}_{q^n}$ where $m|n$ are also described., Comment: 30 pages

The complexity of dual basis of a type I optimal normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$ was determined to be $3n-3$ or $3n-2$ according as $q$ is even or odd, respectively, by Z.-X. Wan and K. Zhou in 2007. We give a new proof to this result by clearly deriving the dual of a type I optimal normal basis with the aid of a lemma on the dual of a polynomial basis., Comment: 7 pages