Single-Error-Correcting Nonbinary Arithmetic Codes

Except for some elementary definitions and fundamentals, the theory of AN code is by and large the theory of binary (radix = 2) arithmetic codes. It is often believed (erroneously) that this theory can be readily generalized to any nonbinary radix. The very fundamental theorems of Brown and Peterson on single-error-correcting codes have been derived for the binary case only. Whereas a generalized version of Brown\´s theorem can be stated and proved relatively easily (as shown here), the one for Peterson\´s theorem is not forthcoming. However, we have succeeded in deriving a theorem for the ternary case (radix = 3) somewhat along the lines of the Peterson\´s theorem as follows. Let denote the smallest positive integer such that the arithmetic weight of in ternary representation is less than . Also ley for some odd prime . Then 3 is a primitive element of if and only if begin{equation} M_3 (A, 3)=(3^{(p-1)/2} + 1)/A. end{equation}