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The residue number system (RNS) is an integer system appropriate for implementing fast digital signal processors since it can support parallel, carry-free, high-speed arithmetic. One of the most important considerations when designing RNS systems is the choice of the moduli set. This is due to the fact that the system's speed, its dynamic range, as well as its hardware complexity depend on both the forms and the number of the chosen moduli. When performing high radix-r(r>2) arithmetic, moduli of forms ra, rb-1 and rc+1 imply simple RNS arithmetic and efficient weighted (radix-r)-to-RNS and RNS-to-weighted (radix-r) conversions. In this paper, new multimoduli high radix-r RNS systems based on moduli of forms ra, rb-1 and rc+1 are presented. These systems will be derived from some recently developed theory. Such systems including moduli of forms ra, rb-1 and rc+1 are appropriate for multiple-valued logic implementations or high radix (r>2) arithmetic using binary logic. The new RNS systems are balanced, achieve fast and simple RNS computations and conversions and implement large dynamic ranges. The specific case of the binary (radix r=2) domain is also presented.