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We propose novel coding and decoding methods for a fully connected K-user Gaussian interference channel. Each transmitter encodes its information into multiple layers and transmits the superposition of those layers. Each receiver employs a constrained partial group decoder (CPGD) that decodes its designated message along with a part of the interference. In particular, each receiver performs a twofold task by first identifying which interferers it should decode and then determining which layers of them should be decoded. Determining the layers to be decoded and decoding them are carried out in a successive manner, where in each step a group of layers with a constraint on its group size is identified and jointly decoded while the remaining layers are treated as Gaussian noise. The decoded layers are then subtracted from the received signal and the same procedure is repeated for the remaining layers. We provide a distributed algorithm, tailored to the nature of the interference channels, that determines the transmission rate at each transmitter based on some optimality measure and also finds the order of the layers to be successively decoded at each receiver. We also consider practical design of a system that employs the quadrature amplitude modulations (QAM) and rateless codes. Numerical results are provided on the achievable sum-rate under the ideal case of Gaussian signaling with random codes as well as on the system throughput under practical modulations and channel codes. The results show that the proposed multi-layer coding scheme with CPGD offers significant performance gain over the traditional un-layered transmission with single-user decoding.