Optimising Codeword Efficiency

Evaluating Usage

Add the usage of the leaf nodes.

If not all of the branches are being used ( $K < 1$ ) for some binary code C, we can shorten them.

Assign the smallest codewords to the most frequent symbols

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Average / Expected Length

$L = \displaystyle\sum_{i=1}^q p_il_i$

Sum the codeword length multiplied by its probability

Average Variance

$V = \displaystyle(\sum_{i=1}^q p_il_i^2 ) - L^2$

A UD-code is minimal if it has minimal length

Minimal

If a binary UD code is minimal

• $l_1 \le l_2 \le … \le l_q$
• The code may be assumed to be instantaneous
• $K = \sum_{i=1}^q 2^{-l_i} = 1$
• $l_{q-1} = l_q$
• $c_{q-1}$ and $c_q$ differ only in their last coordinate