Upper bound of single-track Gray codes and the combined coding method of period 2n

Single-track Gray code is a special class of Gray codes that all the columns of its codewords are shifted equivalent. The upper bound of single-track Gray codes is discussed on necklace orderings and self-dual necklace orderings. As no length n period 2
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single-track Gray code exists, extra reading heads are inevitable to obtain period 2
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code and in the worst situation 2n reading heads may have to be used. To solve this problem, a subdivision code is defined. Combining a single-track Gray code and the subdivision code, a double-track Gray code can be constructed that have period 2
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and may reduce the number of reading heads. In that worst 2n reading heads situation, only n+1 reading heads are adequate when use combined absolute coding method to obtain period 2
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.