Error-correcting coding schemes devised for binary arithmetic are not in general applicable to BCD arithmetic. In this paper, we investigate the new problem of using such coding schemes in BCD systems. We first discuss the general characteristics of arithmetic errors and define the arithmetic weight and distance in BCD systems. We show that the distance is a metric function. Number theory is used to construct a class of single-error-correcting codes for BCD arithmetic. It is shown that the generator of these codes possesses a very simple form and the structure of these codes can be analytically determined.

A new single-error-correcting scheme for binary adders is devised in this paper. A new weight function [called the Arithmetic Series Weight Function (ASWF)] of binary integers is defined. By using this weight function, a simple weight-preserved relation among the operands, sums and carries of binary adders is obtained. The error-correcting scheme is based on this relation. The redundancy of this scheme is comparable to that of the other error-control techniques in binary computer arithmetic. Moreover, the error-correcting procedure is simple and can be carried out by solving a simple algebraic equation involving only additive operations.

A new class of cyclic multiple-error-correcting arithmetic codes is described, and the minimum distance of these codes is determined analytically. A two-step majority-logic decoding scheme is developed for these arithmetic codes. The new class of codes is extended to a new class of arithmetic codes having a primitive cyclotomic factor as the generator and anL-step majority-logic decoding scheme is developed for the extended class of codes.