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We treat the problem of bounding components of the possible distance distributions of codes given the knowledge of their size and possibly minimum distance. Using the Beckner inequality from harmonic analysis, we derive upper bounds on distance distribution components which are sometimes better than earlier ones due to Ashikhmin, Barg, and Litsyn. We use an alternative approach to derive upper bounds on distance distributions in linear codes. As an application of the suggested estimates we get an upper bound on the undetected error probability for an arbitrary code of given size. We also use the new bounds to derive better upper estimates on the covering radius, as well as a lower bound on the error-probability threshold, as a function of the code's size and minimum distance.