Arithmetic error codes in residue number systems with magnitude index

This paper reconsiders the idea of adding a magnitude index to the tesidue representation of numbers. While in the previous papers the magnitude index was intended to be a mean to simplify the non-modular operations, it will be shown that the redundancy implied by the use of a magnitude index can also provide error detection or correction. In order to obtain this result, the range of a Residue Number System is supposed to the divided into intervals of equal width and the residue index of a given number has been defined as an integer locating the number into precisely one interval. In the hypothesis that the magnitude index is given a residue enconding,the conditions under which single residue digit error affecting the given number, or its magnitude index, are stated. It is observed that the redundancy requirements for error detection or correction are the same as for Redundant Residue Number Systems or Product Codes in Residue Number Systems. As an alternate approach, it is shown that the codes under consideration allow detection of any error affecting the residue representation, provided that the error value exceeds a given threshold, and whenever an error is detected, it is possible to determine an approximation of the correct number. The value of the threshold, as well as the error occurring in the approximation of the correct number, decrease as the redundancy increases.