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Combinatorial designs are closely related to linear codes. In recent year, there are a lot of $t$-designs constructed from certain linear codes. In this paper, we aim to construct $2$-designs from binary three-weight codes. For any binary three-weight code $\mathcal{C}$ with length $n$, let $A_{n}(\mathcal{C})$ be the number of codewords in $\mathcal{C}$ with Hamming weight $n$, then we show that $\mathcal{C}$ holds $2$-designs when $\mathcal{C}$ is projective and $A_{n}(\mathcal{C})=1$. Furthermore, by extending some certain binary projective two-weight codes and basing on the defining set method, we construct two classes of binary projective three-weight codes which are suitable for holding $2$-designs.

Sparse code multiple access (SCMA) building upon orthogonal frequency division multiplexing (OFDM) is a promising wireless technology for supporting massive connectivity in future machine-type communication networks. However, the sensitivity of OFDM to carrier frequency offset (CFO) poses a major challenge because it leads to orthogonality loss and incurs intercarrier interference (ICI). In this paper, we investigate the bit error rate (BER) performance of SCMA-OFDM systems in the presence of CFO over both Gaussian and multipath Rayleigh fading channels. We first model the ICI in SCMA-OFDM as Gaussian variables conditioned on a single channel realization for fading channels. The BER is then evaluated by averaging over all codeword pairs considering the fading statistics. Through simulations, we validate the accuracy of our BER analysis and reveal that there is a significant BER degradation for SCMA-OFDM systems when the normalized CFO exceeds 0.02.

We propose a novel iterative numerical method to solve the three-dimensional inverse obstacle scattering problem of recovering the shape of the obstacle from far-field measurements. To address the inherent ill-posed nature of the inverse problem, we advocate the use of a trained latent representation of surfaces as the generative prior. This prior enjoys excellent expressivity within the given class of shapes, and meanwhile, the latent dimensionality is low, which greatly facilitates the computation. Thus, the admissible manifold of surfaces is realistic and the resulting optimization problem is less ill-posed. We employ the shape derivative to evolve the latent surface representation, by minimizing the loss, and we provide a local convergence analysis of a gradient descent type algorithm to a stationary point of the loss. We present several numerical examples, including also backscattered and phaseless data, to showcase the effectiveness of the proposed algorithm.

The discovery of topological quantum states in two-dimensional (2D) systems is one of the most promising advancements in condensed matter physics. Linear Weyl point (LWP) phonons have been theoretically investigated in some 2D materials. Especially, Jin, Wang, and Xu [Nano Lett. 2018, 18, 12, 7755-7760] proposed in 2018 that the candidates with threefold rotational symmetry at the corners of the hexagonal Brillouin zone can host LWP phonons with a quantized valley Berry phase. However, all the candidates with hexagonal lattices may not host LWP phonons at $K$ ($K'$) high-symmetry points (HSPs). Hence, a more accurate recipe for LWP phonons in 2D is highly required. This work provides an exhaustive list of valley LWP phonons at HSPs in 2D by searching the entire 80 layer groups (LGs). We found that the valley LWP phonons can be obtained at HSPs in 11 of the 80 LGs. Guided by the symmetry analysis, we also contributed to realizing the ideal 2D material with valley LWP phonons. We identified the existence of the valley LWP phonons in eleven 2D material candidates with 11 LGs. This work offers a method to search for valley LWPs in 2D phononic systems and proposes 2D material candidates to obtain the valley LWP phonons.

We propose an alternating direction method of multipliers (ADMM) to solve an optimization problem stemming from inverse lithography. The objective functional of the optimization problem includes three terms: the misfit between the imaging on wafer and the target pattern, the penalty term which ensures the mask is binary and the total variation regularization term. By variable splitting, we introduce an augmented Lagrangian for the original objective functional. In the framework of ADMM method, the optimization problem is divided into several subproblems. Each of the subproblems can be solved efficiently. We give the convergence analysis of the proposed method. Specially, instead of solving the subproblem concerning sigmoid, we solve directly the threshold truncation imaging function which can be solved analytically. We also provide many numerical examples to illustrate the effectiveness of the method.

Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then determining their weight distributions have been interesting in coding theory and cryptography. In this paper, a generic construction for binary linear codes with dimension $m+2$ is presented, then a necessary and sufficient condition for this binary linear code to be minimal is derived. Based on this condition and exponential sums, a new class of minimal binary linear codes violating the Ashikhmin-Barg condition is obtained, and then their weight enumerators are determined.