Near optimal single-track Gray codes

Abstract

Single-track Gray codes are a special class of Gray codes which have advantages over conventional Gray codes in certain quantization and coding applications. The problem of constructing high period single-track Gray codes is considered. Three iterative constructions are given, along with a heuristic method for obtaining good seed-codes. In combination, these yield many families of very high period single-track Gray codes. In particular, for m > 3 , length 'r~ = 2", period 2" -2n codes are obtained. Index Terms-Graycodes, heuristic methods, inequivalent sequences, necklaces, quantization, recursive construction, self-dual sequences 2n 5 P 5 2". Examples of high period codes for small n were given in [9], along with a general construction for single-track Gray resulting in a small angular error.codes which leads to the following:

Computing Binary Combinatorial Gray Codes Via Exhaustive Search With SAT Solvers

“…An n-bead binary necklace is an equivalence class of binary n-tuples under rotation [1]. Necklace-based construction of the codes is proved in [22], [23] to be very successive for small . The methods are not easily adapted to produce the codes with larger than 7 because of a rapid increase of the number of necklaces.…”

Determination of optimal period of absolute encoders with single track cyclic gray code

“…So far, at least one realistic solution has been found for the STCGC with length up to 17, using the Karnaugh map method introduced in Ref. [6], the distinction and Gray code property are satisfied simultaneously. Table 5 lists that when the length is prime, most of the words can be used except for only two short codes, i.e.…”

“…To determine the maximum possible period, or called as optimal period, is a challenge in this area. HILTGEN et al [5] , ETZION and PATERSON [6] , and SCHWARTZ and ETZION [7] put forward theoretical approach to constructing STCGC and the optimal period are given up to length of 15. ZINOVIK et al [8] tackled the exhaustive search for binary combinatorial Gray codes with various spreads up to length of 17.…”

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

“…The principle of parameter choosing in the construction and some examples also introduced in section TIT.IT. UPPER BOUND OF STNGLE-TRACK GRAY CODES The constructions of single-track Gray codes can be classified into necklace ordering and self-dual necklace ordering[6][7]. A survey on single-track Gray code is given in[8 ] and some background concepts will be reviewed.Let W = [wo' wI' ... , w n .j] be a length n word.…”