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我英文很爛 又遇到翻譯 請幫幫我~~ IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 7, NOVEMBER 1999 2383 The Structure of Single-Track Gray Codes Moshe Schwartz and Tuvi Etzion, Senior Member, IEEE Abstract—Single-track Gray codes are cyclic Gray codes with codewords of length n, such that all... show more 我英文很爛

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO.

7, NOVEMBER 1999 2383

The Structure of Single-Track Gray Codes

Moshe Schwartz and Tuvi Etzion, Senior Member, IEEE

Abstract—Single-track Gray codes are cyclic Gray codes with

codewords of length n, such that all the n tracks which correspond

to the n distinct coordinates of the codewords are

cyclic shifts of the first track. We investigate the structure of

such binary codes and show that there is no such code with

2n codewords when n is a power of 2. This implies that the

known codes with 2n

���� 2n codewords, when n is a power of

2, are optimal. This result is then generalized to codes over

GF(p), where p is a prime. A subclass of single-track Gray

codes, called single-track Gray codes with k-spaced heads, is

also defined. All known systematic constructions for single-track

Gray codes result in codes from this subclass. We investigate this

class and show it has a strong connection with two classes of

sequences, the full-order words and the full-order self-dual words.

We present an iterative construction for binary single-track Gray

codes which are asymptotically optimal if an infinite family of

asymptotically optimal seed-codes exists. This construction is

based on an effective way to generate a large set of distinct

necklaces and a merging method for cyclic Gray codes based

on necklaces representatives.

Index Terms—Cyclic Gray codes, feedback shift-register, linear

complexity, necklaces, self-dual sequences, single-track codes.
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