A Study On The Algorithm For Coding The Single—track Gray Codes With D-plet-necklace Structure

Single-track Gray code (STGC) is a special kind of Gray code such that each column of the codeword matrix is the cyclic shift of the first column so that any single track is adequate for encoding the complete set of codeword matrix of any resolution. Therefore, STGC is a suitable alternative for absolute encoders with small size and high resolution. The coding theory of STGCs is discussed in this dissertation, and a novel structure uni-fying all the known codes is proposed and the main contents are shown as follows.Further efficient searching algorithm, leading to the complete STGC solution set of word length6, is developed by proposing and applying three theorems assuring the property of the generating sequence of STGCs. And two novel subclasses of STGCs are found in the solution set.Based on the structure of the conventional STGCs, the basic structure of the first novel subclass found in length6complete solution is presented, which is called twin-necklace ordering STGC. The upper bound of the period and the reading head distribution are also discussed.The method for constructing twin-necklace ordering is proposed from single-track property, mono-difference and distinctness. The compatibility to the first two properties are fully satisfied and the method for constructing the seed code sequences having these two properties is obtained, called’Single-track and Mono-difference Construction’. For distinctness, a necessary condition and a ranking algorithm are proposed to facilitate the searching procedure for distinctness. Therefore, twin-necklace ordering STGCs can be obtained by a construct-searching method.The second novel subclass found in length6complete searching solution is defined as triplet-necklace ordering STGC. Its basic structure, the reading head distribution and the construction method are also proposed based on the discussion of twin-necklace or-dering, and some length9triplet-necklace ordering STGCs are found afterwards. More-over, we define a new kind of STGC as d-plet-necklace ordering where the subcycle of their reading head interval is d. The construction method of d-plet-necklace order-ing STGCs is obtained, and some length8quadruplet-necklace(d=4) and length10quintuplet-necklace (d=5) ordering STGCs are found.All the single-track Gray codes found so far are all subclasses of d-plet-necklace ordering STGCs, and can be reclassified by different value of d, where d is any integer factor of n. The conventional necklace ordering is d-plet-necklace ordering with d=1, the self-dual necklace ordering is d-plet-necklace ordering with d=n and other known STGCs are d-plet-necklace ordering with1<d<n which are called multiple-strand-necklace orderings.The implementation of a confirmatory single-track absolute encoder prototype using a length11period2046STGC is introduced at the end of the dissertation, including mechanical structure, coding disc and circuit design.