## 1 Introduction  
Due to its capability for global seamless coverage, satellite communication has been extensively studied in recent years [1]. Since 2017, the 3 rd Generation Partnership Project (3 GPP) has been exploring the integration of satellite communication with ground cellular systems for air-space integration technology, proposing technical standards to support non-terrestrial networks (NTNs) [2, 3]. To achieve optimal spatial coverage, the radiation pattern of the array antenna for satellite communication should meet specific requirements, such as having a high gain, a narrow 3 dB beamwidth, and low sidelobe levels [4, 5].  
The synthesis of the array pattern, which can direct the main lobe to the desired direction with appropriate beamwidth and low sidelobe levels in other directions, has been widely studied [6]. However, as the number of array elements increases, the feeding network with TR-modules of a large-scale array becomes very complex and expensive [7]. A thinned array, created by removing some elements from a periodic array, offers a promising solution for large-scale arrays by maintaining low sidelobe levels and simplifying the feeding network [8]. In the textbook [6], methods such as Schelkunoff polynomial, Fourier transform, Woodward-Lawson, Taylor, and back-projection are proposed to achieve the desired array radiation pattern. However, due to the complexity of the desired pattern,such as multiple beam directions and stringent sidelobe level requirements, these methods often fall short of achieving the desired results. Also, the adaptively thinned [8], deterministic binary sequences [9], probability density tapering [10], modified iterative Fourier technique [11, 12], precoded subarray structures [13], and perturbational method [14] are also proposed to synthesis of the thinned array.  
Additionally, the nonlinear optimization methods derived from biological behavior [15, 16, 17, 18, 19, 20, 21, 22], such as differential evolution (DE) [16, 17], particle swarm optimization (PSO) [18], simulated annealing (SA) [19], invasive weed optimization (IWO) [20], and genetic algorithms (GAs) [21, 22], are suitable for solving the nonlinear problems of the synthesis of the thinned array. These algorithms usually iteratively reduce the gap (objective function) between the desired and current radiation patterns. However, with large array scales, nonlinear optimization methods can be time-consuming and may struggle to converge. Furthermore, the neural network method is also applied to achieve the synthesis of the thinned array [23, 24]. To realize superior performance, extensive data should be provided to train the neural network.  
To achieve the synthesis of the large-scale thinned array, branch and bound technology with convex optimization is proposed and studied in this paper. Branch and bound technology can decompose optimization into several subproblems. The convex optimization can transform the optimized variable binary [0, 1] of the thinned array into the continues variable, i.e., from 0 to 1. Ultimately, the synthesis of the 64 \(\times\) 64 thinned array with the desired radiation pattern is achieved. The simulated and measured results of a thinned array are presented to verify the results.