Error-Correcting Codes in Binary-Coded Radix-r Arithmetic

Abstract

This paper presents a new class of AN codes of high efficiency capable of correcting single errors in radix-r arithmetic with the binary coded digits—BCr—system. The errors corrected within a single radix-r digit are single errors of the binary digits with weight ±w <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ≤ r–1,i=1,2,···, m, which are used to encode the BCr digits. The corresponding arithmetic unit is made out of slices, one for each BCr digit. The AN codes considered have a generator A of the form A = τ·p where τ, p odd and rp=n-1 ≤ τ < p. The paper establishes the selection criteria of r and p such that the code range of the AN codes is equal to M = r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> ± 1. The criteria are applied to the BCD system, and we determine all τ < 100 and p < 200 for the most important BCD codes with weights ± wi < r –1, i = 1, 2, 3, 4. For each BCD code, the paper gives the numbers τ < 100 for which AN codes exist, and the maximum code efficiency E = 2·m·k/(A-1) attained at some p < 200.