Title: FleXOR: Trainable Fractional Quantization. Abstract: Parameter quantization is a popular model compression technique due to its regular form and high compression ratio. In particular, quantization based on binary codes is gaining attention because each quantized bit can be directly utilized for computations without dequantization using look-up tables. Previous attempts, however, only allow for integer numbers of quantization bits, which ends up restricting the search space for compression ratio and accuracy. Moreover, quantization bits are usually obtained by minimizing quantization loss in a local manner that does not directly correspond to minimizing the loss function. In this paper, we propose an encryption algorithm/architecture to compress quantized weights in order to achieve fractional numbers of bits per weight and new compression configurations further optimize accuracy/compression trade-offs. Decryption is implemented using XOR gates added into the neural network model and described as $\tanh(x)$, which enable gradient calculations superior to the straight-through gradient method. We perform experiments using MNIST, CIFAR-10, and ImageNet to show that inserting XOR gates learns quantization/encrypted bit decisions through training and obtains high accuracy even for fractional sub 1-bit weights.