Entropy lower bound for prefix codes

Every binary prefix code has expected length at least its source entropy; Shannon lengths achieve expected length strictly below $H+1$.

For a discrete memoryless source of entropy $H$, every uniquely decodable binary code has expected length at least $H$. Shannon code lengths satisfy $H\le \overline{\ell }<H+1$, and block coding can make the expected length per source symbol arbitrarily close to $H$.