Optimal normalizing coding theory and methods for Counter-Meshing Gears mechanisms

The counter-Meshing Gears (CMG) mechanism is a discrimination mechanism which can be used in combination locks for high-consequence system surety. For an arbitrary binary Unlocking Symbol Sequence composed of equal number symbols of A and B, i.e. NA+NB, how to realize the binary discrimination teeth coding of its corresponding CMG mechanism with minimum fixed gear levels C and gear divisions D, is an important practical problem which is firstly well defined as the Optimal Normalizing CMG Coding Problem. With the toolbox comprising previously reported terms and methods, e.g. the CMG classification method, the 2-D Maze Map and the 3-color circular alternant coloring method for Critical Trap Grids (CTGs), optimal normalizing coding theory and methods for CMG Mechanisms are systematically discussed. Two optional coding methods, and their minimum requirement for the coding space (characterized with CD) and coding algorithm, are all presented. A Figure of Merit (FoM) which characterizes the CMG coding efficiency is defined on the coding space and the symbol length of the Unlocking Symbol Sequence which dedicated for. By the FoM with clear physical meanings, the two optional Optimal Normalizing CMG Coding methods are compared, and it is concluded that the first type CMG mechanism with a coding space of C=3 and D=N+2 is the preferred method. As to the first type CMG mechanism, there is no difference between the Optimal Normalizing Coding, the previously reported Optimized coding with minimum gear levels, thus the minimum coding space of C=3 and D=N+2 are both needed and the 3-color circular alternant CTGs coloring method is a suitable coding method for both two. With application of the 3-color circular alternant CTGs coloring method, a distinct fingerprint feature can be revealed in the 2-D verification maze map that all CTGs are circular regularly allotted to only three color sets, i.e. the predesigned gear-teeth meshing between the two coupled composite gear A and B for error-locking function will alternately happen in only three discrimination gear levels.