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A Simple Recursive Shannon Code


The Shannon code is a very simple suboptimal scheme for computing prefix-free codelengths for a given memoryless source. A recursive version of the well-known Shannon code (RSh) is investigated in this paper. It has a redundancy which never exceeds that of the Shannon code. Unlike the Huffman code, the RSh code and its variations can be directly applied to sources with an infinitely countable alphabet. The average redundancy, taken over the set of all $n$-tuple distributions, is considered as a criterion for code comparisons. For $n>40$, the average redundancy is approximately 0.14 bits for the RSh code, compared to approximately 0.5 and 0.03 bits for the Shannon and Huffman codes, respectively. For large $n$, a constrained version of the RSh code, called CRSh, has an average redundancy of only 0.07 bits for unsorted symbol probabilities. If the symbol probabilities are sorted in descending order, then a very simple modification of the RSh algorithm results in a code which is near-optimal from the average redundancy point of view.

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