Uncertainty principle plays an important role in physics, mathematics, signal processing and et al. In this paper, based on the definition and properties of discrete linear canonical transform (DLCT), we introduced the discrete HausdorffYoung inequality. Furthermore, the generalized discrete Shannon entropic uncertainty relation and discrete Rényi entropic uncertainty relation were explored. In addition, the condition of equality via Lagrange optimization was developed, which shows that if the two conjugate variables have constant amplitudes that are the inverse of the square root of numbers of non-zero elements, then the uncertainty relations touch their lowest bounds. On one hand, these new uncertainty relations enrich the ensemble of uncertainty principles, and on the other hand, these derived bounds yield new understanding of discrete signals in new transform domain.