Single-Error-Correcting Nonbinary Arithmetic Codes

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Abstract
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. LetM_3 (A, d)denote the smallest positive integer such that the arithmetic weight ofA M_3 (A, d)in ternary representation is less thand. Also leyA = 2pfor some odd primep. Then 3 is a primitive element ofGF(p)if and only if begin{equation} M_3 (A, 3)=(3^{(p-1)/2} + 1)/A. end{equation}

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