An Approximation Algorithm for Constructing Error Detecting Prefix Codes

A k-bit Hamming prefix code is a binary code with the following property: for any codeword x and any prefix y of another codeword, both x and y having the same length, the Hamming distance between x and y is at least k. Given an alphabet A = [a1, . . . , an] with corresponding probabilities [p1, . . . , pn], the k-bit Hamming prefix code problem is to find a k-bit Hamming prefix code for A with minimum average codeword length ∑ n i=1 pili, where li is the length of the codeword assigned to ai. In this paper, we propose an approximation algorithm for the 2-bit Hamming prefix code problem. Our algorithm spends O(n log n) time to calculate a 2-bit Hamming prefix code with an additive error of at most O(log log log n) bits with respect to the entropy H = − ∑ n i=1 pi log2 pi.