## 1 Introduction  
Optical communication systems are usually deployed over fiber-optic links [1; 2; 3], free-space optical (FSO) links [4; 5; 6], or hybrid FSO-fiber links [7]. Recent advances in photonics integrated circuits (PIC) have been greatly pushing forward the worldwide application of optical communications [8; 9; 10]. Although they have enabled a capacity-approaching communication [11; 12; 13], fiber-optic links may be too fragile or costly to be deployed in some environments, e.g., seismic belts. As a result, FSO links are more favorable due to their easy and fast communication link reconstruction. Although the channel loss of FSO links is not stable, and typically higher than that of fiber-optic links, FSO communication systems provide free-scalable channels for spatial mode division (SDM) multiplexing, e.g., orbital angular momentum (OAM) multiplexing [14; 15; 16; 17]. Despite their free-scalable characteristics, spatially multiplexed modes hardly preserve their orthogonality when transmitting over the atmospheric FSO links, resulting in dynamic inter-mode crosstalk [18; 19; 20]. There are several ways to mitigate the inter-mode crosstalk, including wavefront sensor (WFS) or wavefront sensorless based adaptive optics (AO) systems [21; 22; 23], digital multi-input multi-output (MIMO) equalization [24; 25; 26], and advanced forward error correction (FEC) based channel coding techniques [27; 28; 29]. AO systems are usually implemented in satellite communications to mitigate wavefront distortion, while their commercial application in near-Earth FSO links is greatly limited by the expensive WFS and deformable mirror (DM). In some instances, a wavefront sensorless AO is used as a trade-off between cost and reliability. MIMO equalization is also often used to relieve inter-mode crosstalk among the multiplexed spatial modes. It is, however, preferentially used in the FSO links affected by weak-to-medium atmospheric turbulence. When spatial modes are transmitted in a strong atmospheric turbulence environment, the unwanted inter-mode crosstalk is not only limited to adjacent spatial modes, but also spread widely across other spatial modes. As a result, the use of more mode detectors is expected at the receiver side to capture the distorted signals in correlated modes, followed by more computationally complex MIMO equalization. If the quantity of mode detector is not sufficient, MIMO equalization may fail to work due to data loss. FEC based channel coding techniques have been fully developed over decades, and extensively used in error-free digital fiber-optic communication systems. However, FEC coding-only solution can't guarantee a reliable data transmission over long reach FSO links, especially in the strong atmospheric turbulence environment. In this paper, we discuss the high-speed OAM multiplexed FSO communication systems, enabled by AO, low-density parity-check (LDPC) coding, spatial offset (SO), and joint Huffman and LDPC coding. First, we experimentally study an AO assisted, LDPC coded, OAM-based FSO communication system. Briefly, four OAM multiplexed mode channels that in total carry 500 Gbps quadrature phase shift keying (QPSK) signals are transmitted over wavelength-division multiplexed (WDM) channels with 50 GHz spacing. The turbulence-induced inter-mode crosstalk is compensated by a wavefront sensorless AO setup. Subsequently, error-free communication can be achieved with a strong LDPC coding scheme. The minimum coding gain of 3.9 dB is achieved at BER = \(2\times 10^{-2}\) for OAM states \(\pm 2\) and \(\pm 6\). Second, we present another inter-mode crosstalk mitigation solution in an atmospheric turbulence limited OAM multiplexed FSO links. The OAM mode crosstalk is first relieved by optical spatial mode offset, and then further resolved by coded modulation technology. Huffman coding and optimal constellation design techniques are applied to generate the quadrature amplitude modulation (QAM) formats, i.e., 5/9-QAM formats.