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Kraft Inequality for UD VLCs

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Theorem

Let C\mathcal{C} be a UD DD-ary n-th order VLC for a discrete source {Xi}i=1\{X_i\}_{i=1}^\infty with alphabet X\mathcal{X} and let l1,,lM\mathscr{l}_{1},\cdots,\mathscr{l}_{M} be the lengths of the code’s M=XMM=|\mathcal{X}|^M codewords. Then these codeword lengths satisfy the Kraft Inequality with base DD i=1MDli1\sum\limits_{i=1}^{M}D^{-\mathscr{l}_{i}}\le1