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
M_3 (A, d)
denote the smallest positive integer such that the arithmetic weight of
A M_3 (A, d)
in ternary representation is less than
d
. Also ley
A = 2p
for some odd prime
p
. Then 3 is a primitive element of
GF(p)
if and only if \begin{equation} M_3 (A, 3)=(3^{(p-1)/2} + 1)/A. \end{equation}