Bounds on the redundancy of Huffman codes (Corresp.)

New upper bounds on the redundancy of Huffman codes are provided. A bound that for
2/9 \leq P_{1} \leq 0.4
is sharper than the bound of Gallager, when the probability of the most likely source letter
P_{1}
is the only known probability is presented. The improved bound is the tightest possible for
1/3 \leq P_{1} \leq 0.4
. Upper bounds are presented on the redundancy of Huffman codes when the extreme probabilities
P_{1}
and
P_{N}
are known.