Cyclic and multiresidue codes for arithmetic operations

In this paper, the cyclic nature of codes is defined after a brief summary of previous work in this area is given. New results are shown in the determination of the range for single-error-correcting codes when is the product of two odd primes and , given the orders of 2 modulo and modulo . The second part of the paper treats a more practical class of arithmetic codes known as separate codes. A generalized separate code, called a multiresidue code, is one in which a number is represented as begin{equation} [N, mid N mid _ {m1}, mid N mid _{m2}, cdots , mid N mid _{mk}] end{equation} where are pairwise relatively prime integers. For each code, where is composite, a multiresidue code can be derived having error-correction properties analogous to those of the code. Under certain natural constraints, multiresidue codes of large distance and large range (i.e., large values of ) can be implemented. This leads to possible realization of practical single and/or multiple-error-correcting arithmetic units.