Spiking neural networks (SNNs) are mimicking computationally powerful biologically inspired models in which neurons communicate through sequences of spikes, regarded here as sparse binary sequences of zeros and ones. In neuroscience it is conjectured that time encoding, where the information is carried by the temporal position of spikes, is playing a crucial role at least in some parts of the brain where estimation of the spiking rate with a large latency cannot take place. Motivated by the efficiency of temporal coding, compared with the widely used rate coding, the goal of this paper is to develop and train an energy-efficient time-coded deep spiking neural network system. To ensure that the similarity among input stimuli is translated into a correlation of the spike sequences, we introduce correlative temporal encoding (CTE) and extended correlative temporal encoding (ECTE) techniques to map analog input information into input spike patterns. Importantly, we propose an implementation where all multiplications in the system are replaced with at most a few additions. As a more efficient alternative to both rate-coded SNNs and artificial neural networks (ANNs), such system rep-resents a preferable solution for the implementation of neuromorphic hardware. We consider data classification tasks where input spike patterns are presented to a feed-forward architecture with leaky-integrate-and-fire (LIF) neurons. The SNN is trained by backpropagation through time with the objective to match sequences of output spikes with those of specifically designed target spike patterns, each corresponding to exactly one class. During inference the target spike pattern with the smallest van Rossum distance from the output spike pattern determines the class. Extensive simulations indicate that the proposed system achieves a classification accuracy at par with that of state-of-the-art machine learning models.