Single-track Gray codes with non-k-spaced heads

Single-track Gray codes are cyclic Gray codes with an extra property that every column of all the codewords is a cyclic shift of the first column. A subclass of single-track Gray codes with non-k-spaced heads is introduced, which does not exist for the length that is lower than 6 and can not be obtained by any transformation from the known k-spaced codes. A basic construction for one category of such non-k-spaced heads singletrack Gray codes with a subcycle of 2 is given, and the rules of choosing seed codes are discussed. Moreover, the conditions that different subclasses of single-track Gray codes can transform into each other are also introduced.