On the redundancy of optimal codes with limited word length

Some limitations are given on the redundancy of D-ary codes with maximal codeword length L. An upper bound that improves on a previous result of E.N. Gilbert (1971) is given. It is then shown that the redundancy of these constrained codes is very close to that of the unconstrained Huffman codes when the number of codewords N is such that ND/sup 1-L/ becomes negligible. A tight bound is given on the redundancy when only the most likely probabilities are known. In the binary case, a tight lower bound is given on the redundancy when only the least likely probability is known.<>