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SPECIAL PROPERTIES OF HAMMING CODES THAT APPEAR WHEN SYNTHESIZING SELF-CHECKING DIGITAL DEVICES

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Several properties of classical Hamming codes associated with the features of detecting errors that occur simultaneously in both information and control bits of code words are revealed, which is relevant in many applications, for example, in the synthesis of self-checking and fault-tolerant digital computing devices and systems. It is shown that when the number of cases of information bits m tends to infinity, the number of undetectable errors that arrose simultaneously in the information and control bits of Hamming codes increases by 2k–1 times with k being the number of control bits, as compared to the number of undetectable errors that occur only in the information bits of their code words. It is also proved that for m→∞, the ratio of the number of undetectable errors that occur simultaneously in information and control bits to the number of undetectable errors appearing only in information bits for a Hamming code with k + q control bits increases by 2q times with respect to the ratio for the Hamming code with k control bits (here q is an arbitrary natural number). The established characteristics of Hamming codes should be taken into account when synthesizing self-checking digital devices. Examples are given of the synthesis of built-in control circuits using the Boolean complement method with consideration of the revealed properties of Hamming codes.

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