The Check Positions of Hamming Codes and the Construction of a 2EC-AUED Code

How Guan Aun ; Ang Miin Huey

Bulletin of the Malaysian Mathematical Society, Tome 21 (1998) no. 2

Résumé

Hamming Code is the oldest and the most commonly used single error correcting and double errors detecting code. For implication, it is constructed over the field GF(2). For each there is a . Hamming Code where and A message word of length k is encoded using a generating matrix G into a codeword of length n . This amounts to inserting r parity check digits into the message word. The positions of the parity check digits in the codeword are called the check positions of the code (with respect to G ). A received word is then decoded using a parity check matrix H . If the check positions of the code are in the coordinates of the codeword, then the rows of H are called the check rows of the code. We proved in this paper that for any parity check matrix of a Hamming Code, there exists a generating matrix G of the code, such that the check rows of the code are linearly independent. We believe that this fact is contained implicitly in a paper of Hamming [7] but we cannot find any explicit proof in existing literature. Using the above fact, we construct a 2 EC-AUED code.