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Building on deep representation learning, deep supervised hashing has achieved promising performance in tasks like similarity retrieval. However, conventional code balance constraints (i.e., bit balance and bit uncorrelation) imposed on avoiding overfitting and improving hash code quality are unsuitable for deep supervised hashing owing to their inefficiency and impracticality of simultaneously learning deep data representations and hash functions. To address this issue, we propose probabilistic code balance constraints on deep supervised hashing to force each hash code to conform to a discrete uniform distribution. Accordingly, a Wasserstein regularizer aligns the distribution of generated hash codes to a uniform distribution. Theoretical analyses reveal that the proposed constraints form a general deep hashing framework for both bit balance and bit uncorrelation and maximizing the mutual information between data input and their corresponding hash codes. Extensive empirical analyses on two benchmark datasets further demonstrate the enhancement of compactness and informativeness of hash codes for deep supervised hash to improve retrieval performance (code available at: https://github.com/mumuxi/dshwr).