Parity codes

Abstract

Motivated by a problem posed by Hamming in 1980, we define even codes. They are Huffman type prefix codes with the additional property of being able to detect the occurrence of an odd number of 1 bit errors i the message. We characterize optimal even codes and describe a simple method for constructing the optimal codes. Further, we compare optimal even codes with Huffman codes for equal frequencies. We show thet the maximum encoding in an optmal even code is at most two bits larger than the maximum encoding. in a Hauffman tree. Moreover, it is always possible to choose an optimal even code such that this difference drops to 1 bit. Finally, we compare average sizes. We show that the average size of an encoding in a optimal even tree is at least 1/3 and at most 1/2 of a bit larger than that of a Huffman tree. These values represent the overhead in the encoding sizes for having the hability to detect an odd number of errors in the message.