A note on the construction of error detecting/correcting prefix codes

Abstract

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 = [ a 1 , … , a n ] with corresponding probabilities [ p 1 , … , p n ] , the k-bit Hamming prefix code problem is to find a k-bit Hamming prefix code for A with minimum average codeword length ∑ i = 1 n p i ℓ i , where ℓ i is the length of the codeword assigned to a i . In this paper, we propose a general approximation algorithm for the k-bit Hamming prefix code problem. Let α k be an O ( r k ( n ) ) -time algorithm for calculating fixed-length codes with Hamming distances k whose codewords are d k ( n ) bits longer than ⌈ log 2 n ⌉ . Our algorithm uses α k to calculate a k-bit Hamming prefix code in O ( r k ( n ) + n log n ) time with an additive error of at most O ( d k ( n ) + log ∗ n ) bits with respect to the optimal prefix code for A, under reasonable assumptions on the function d k .