Non-Separate Arithmetic Codes

It is shown that a non-separate arithmetic code that preserves both addition and multiplication must be an AN code where the generator A is an idempotent element of the ring being used. An idempotent element is one that satisfies the equation x squared = x. Given this type of code, its ability to detect errors in arithmetic expressions is explored and shown to be poor, due to error masking in multipliers. The constraints placed on a non-separate multiplication-preserving arithmetic code that avoids such problems are discussed. The simplest code satisfying these conditions turns out to be an AN+B code where both A and B are idempotent elements. Conditions for the existence of this type of code are given along with a list of examples. The fault tolerance provided by these codes is then considered for a specific example. This paper considers an approach and addresses the problem of classification of error- detecting codes than can protect large arithmetic expressions involving addition and multiplication, or more correctly circuits that implement such expression.