Cyclic and multiresidue codes for arithmetic operations
T.R.N. Rao, Óscar García · IEEE Transactions on Information Theory · 1971
In this paper, the cyclic nature ofANcodes 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-correctingANcodes whenAis the product of two odd primesp_1andp_2, given the orders of 2 modulop_1and modulop_2. 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 numberNis represented as \begin{equation} [N, \mid N \mid _ {m1}, \mid N \mid _{m2}, \cdots , \mid N \mid _{mk}] \end{equation} wherem_iare pairwise relatively prime integers. For eachANcode, whereAis composite, a multiresidue code can be derived having error-correction properties analogous to those of theANcode. Under certain natural constraints, multiresidue codes of large distance and large range (i.e., large values ofN) can be implemented. This leads to possible realization of practical single and/or multiple-error-correcting arithmetic units.