Correcting codes for arithmetic errors

Abstract

In computers with a built-in logical check the main difficulty in correcting the consequences of random malfunctioning is to return to a correctly executed part of the program. In some cases, such as when the malfunctioning occurs in the arithmetic unit, a simple repetition of the operation is carried out. At the same time, it is not always easy to restore the spoiled storage location, and corresponding programs for analysis and resetting are exceptionally complicated, particularly if multi-programming is allowed for. One method of simplifying the resetting problem is to construct codes which make it possible to correct the malfunctioning at least in one digit. Codes of this kind can also be considered as a step on the way to the automation of fault detection. In the accepted terminology, codes which enable errors to be corrected and on which arithmetic operations can perform are called arithmetic error correcting codes, although as is clear from the previous paragraph, this term does not accurately reflect the possible region of application of these codes. This work is not the first in which arithmetic error correcting codes have been examined. Here we must note, first, the work of Brown [l]. which gives a detailed analysis of the code of the form An + B, where A and B are constants and n the number to be coded. However, Brown's statement of the problem, i.e. the search for codes of a preset form, cannot be said to be satisfactory, since this class is to some extent artificial. Its choice is restricted by the possibility of executing the addition operation only (with subsequent subtraction of a constant). The operation of multiplying two codes of the given form is thus not defined. In order to present the problem correctly we have, first of all, to remember that because of the fact that the adder is finite, the computer unit operates not with an infinite set of numbers (integers or rational numbers), but with a finite set of residues modulo m. ∗ ∗ All algebraic terms and concepts occurring in the text are introduced in “Modern Algebra”, Parts 1 and 2, by Van der Waerden. Below we shall write the modulo m in the form of a product of powers of simple factors m = p 1 α 1 p 2 α 2 h. p k α k , P i ≠ P j , for i ≠ j. Suppose that the ring of numbers with which the adder operates is the ring of residues modulo m = 2 n . In this ring the distance between codes (in their binary representation) defined in the normal way is greater than or equal to one, i.e. there are codes which differ only in a single digit. The most natural method of increasing the distance between codes, clearly, consists in the construction of the ring of residues modulo m 1 > m such that the initial ring is isomorphically imbedded in it, the distance between codes of the isomorphic image of the initial ring in the constructed ring now greater than or equal to the desired distance. Let us prove that such an imbedding is possible.