In laser-based free-space optical communication, besides OAM beams, Hermite-Gaussian (HG) modes or HG-mode coherently composed states (HG-MCCS) can also be adopted as the information carrier to extend the channel capacity with the spatial pattern based encoding and decoding link. The light field of HG-MCCS is mainly determined by three independent parameters, including indexes of HG modes, relative initial phases between two eigenmodes, and scale coefficients of the eigenmodes, which can obtain a large number of effective coding modes at a low mode order. The beam intensity distributions of the HG-MCCSs have obvious distinguishable spatial characteristics and can keep propagation invariance, which are convenient to be decoded by the convolutional neural network (CNN) based image recognition method. We experimentally utilize HG-MCCS to realize a communication link including encoding, transmission under atmospheric turbulence (AT), and decoding based on CNN. With the index order of eigenmodes within six, 125 HG-MCCS are generated and used for information encoding, and the average recognition accuracy reached 99.5% for non-AT conditions. For the 125-level color images transmission, the error rate of the system is less than 1.8% even under the weak AT condition. Our work provides a useful basis for the future combination of dense data communication and artificial intelligence technology.