Hamming Codes with Check Bits that are Described by Self-Dual and Self-Quasidual Boolean Functions
Abstract
The authors of the paper investigate the diagnostic features of Hamming codes. The analysis showed that for specific values of the number of bits in the codewords of classical Hamming codes, the check bits are described by self-dual Boolean functions. The authors have established a pattern in which for a certain number of data bits, the check bits of Hamming codes can be calculated by self-dual Boolean functions, and for a certain number by self-quasidual Boolean functions. The encoders of such Hamming codes will be self-dual devices. The encoders of the first codes will be self-dual devices, and the second ones will be self-quasidual devices, respectively. In addition, the same conditions holds true when using its relatively modified Hamming code, for which another check is additionally defined in the form of a parity bit for all data and check bits of the classical Hamming code (extended Hamming code). These features can be effectively used in the synthesis of digital devices and systems with the detection of faults and errors in calculations. This requires the use of a pulse mode of operation and the introduction of temporary redundancy in the operation of the circuits. This makes it possible to use classical and extended Hamming codes in the synthesis of self-checking and fault-tolerant devices with checking of the sign of self-duality and self-quasiduality of calculated functions.