V.E. Vickers and J. Silverman, A technique for generating specialized Gray codes, IEEE Trans. Comput. C-29(1980) 329--331.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....n = 4 0 00 000 0000 1010 1 01 001 0001 1011 11 011 0011 1001 10 111 0111 1101 101 1111 0101 100 1110 0100 110 1100 0110 010 1000 0010 Figure 1: Examples of 1, 2, 3, and 4 bit antipodal Gray codes. are: restricting where bit ips can occur [BR96] requiring the same number of ips on any given bit [VS80, RC81]; enforcing non locality conditions [Ram90] maximizing the gap, i.e. the shortest maximal consecutive sequence of 0 s (or 1 s) among all bit positions [GLN88] and requiring certain monotonicity properties [SW95] But this is just a small sample. For a survey of Gray codes see [Sav97] and for a ....

V. E. Vickers and J. Silverman. A technique for generating specialized Gray codes. IEEE Trans. Comput., 29(4):329-331, 1980. 16

....2. Question. How large can the digirth of the digraph of a Gray code be Attention has so far focused on sparse graphs the fewer edges, the more restrictions. Shouldn t typical Gray codes induce many edges The largest code we know of which induces a complete graph is an 8 bit code appearing in [10]. Problem. Construct n bit Gray codes which induce Kn . Acknowledgments In 1989 and 1990, the rst author attended the University of Minnesota, Duluth Research Experiences for Undergraduates program, supported by the National Science Foundation (DMS 9000742) and the National Security Agency ....

V.E. Vickers and J. Silverman, A technique for generating specialized Gray codes, IEEE Trans. Comput. C-29(1980) 329-331.

....the induced digraph of a Gray code be Attention has so far focused on sparse graphs the fewer edges, the more restrictions on coding. Furthermore, many inductive constructions of Gray codes omit edges. It seems likely, however, that typical Gray codes induce many edges. Vickers and Silverman [14] give a balanced cyclic code on 8 bits which induces a complete graph (and whose induced digraph contains every directed edge) this is the largest such example of which we are aware. Question. Construct Gray codes on n bits which induce Kn or, even better, whose induced digraphs contain ....

Vickers, Virgil E. and Jerry Silverman. A technique for generating specialized Gray codes. IEEE Transactions on Computers C-29 (1980), 329--331.

....whereas the highest order bit changes only twice, counting the return to the first element. In certain applications, it is necessary that the number of bit changes be more uniformly distributed among the bit positions, i.e. a balanced Gray code is required. See Figure 1(b) for an example from [VS80]. Uniformly balanced Gray codes were shown to exist for n a power of two by Wagner and West [WW91] For general n, balancing heuristics were suggested, but not proved, in [LS81, VS80, RC81] Recently we have shown, using the Robinson Cohn construction [RC81] that balanced Gray codes exist for ....

....distributed among the bit positions, i.e. a balanced Gray code is required. See Figure 1(b) for an example from [VS80] Uniformly balanced Gray codes were shown to exist for n a power of two by Wagner and West [WW91] For general n, balancing heuristics were suggested, but not proved, in [LS81, VS80, RC81]. Recently we have shown, using the Robinson Cohn construction [RC81] that balanced Gray codes exist for all n in the following sense: Let a = b2 n =nc or b2 n =nc Gamma 1, so that a is even. For each n 1 there is a cyclic n bit Gray code in which each bit position changes either a or a ....

V. E. Vickers and J. Silverman. A technique for generating specialized Gray codes. IEEE Transactions on Computers, C-29:329--331, 1980.

....problem involving digitization of analogue data. Since # Supported in part by NSF grant DMS9302505 1 the electronic journal of combinatorics 3 (1996) #R25 2 then, binary Gray codes have been used in a wide variety of other applications including databases,experimentaldesign,andpuzzlesolving[4,5,6,7,8]. As discussed, for example, in [7] the BRGC scheme, though su#cient to solve the communications problem, is not adequate for certain other applications because of its lack of uniformity . The term uniformity refers to the manner in which the bits change in the Gray code. Several di#erent ....

....The term uniformity refers to the manner in which the bits change in the Gray code. Several di#erent measures of uniformity and techniques to construct Gray codes satisfying these measures have been proposed in literature. Two such measures are the distribution of transition counts [2, 5, 6, 8] and the gap [7] of a code. Gray codes which are uniform with respect to the former measure are referred to as ba l anced Gray codes. To make this notion precise, associate with an n bit cyclic Gray code L n = w 1 ,w 2 , w 2 n , the transition sequence of bit positions s = s 1 ,s 2 , s 2 ....

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V. E. Vickers and J. Silverman, A Technique for Generating Specialized Gray Codes, IEEE Trans. Comput. C-29 (1980) 329-331.

....bit changes 2 n Gamma1 times, whereas the highest order bit changes only twice, counting the return to the first element. A balanced Gray code, in which the 2 n bit changes are distributed as equally as possible among the n bit positions, has long been sought and heuristics have been proposed [15, 19]. Only in the case where n is a power of two is a balanced Gray code known to exist [18] In other applications, the requirement is to maximize the shortest maximal consecutive sequence of zeroes (or ones) among all bit positions [8] Here we consider a new constraint. Define the weight of a ....

V. E. Vickers and J. Silverman, A technique for generating specialized Gray codes, IEEE Transactions on Computers C-29 (1980), 329-331.

....binary vectors of length n such that each pair of adjacent vectors (including the first and last) differ in a single position. There is extensive literature, going back at least to 1872 see for example [1] 2] 4] 6] 9] 12] 16] 17] 19] 21] 23] 29] 38] 40] 47] 50] 51] [54], 55] 61] 62] As we will show, the classical version is the special case G = 1 n of the following. Theorem. Let G be a finite group generated by reflections R 1 , R n . Then there is a Hamiltonian circuit in the Cayley diagram for G corresponding to these generators. In other words ....

V. E. Vickers and J. Silverman, A technique for generating specialized Gray codes, IEEE Trans. Computers, 29 (1980), 329-331. - 22 -

....problem involving digitization of analogue data. Since Supported in part by NSF grant DMS9302505 the electronic journal of combinatorics 3 (1996) #R25 2 then, binary Gray codes have been used in a wide variety of other applications including databases, experimental design, and puzzle solving [4, 5, 6, 7, 8]. As discussed, for example, in [7] the BRGC scheme, though sufficient to solve the communications problem, is not adequate for certain other applications because of its lack of uniformity . The term uniformity refers to the manner in which the bits change in the Gray code. Several different ....

....The term uniformity refers to the manner in which the bits change in the Gray code. Several different measures of uniformity and techniques to construct Gray codes satisfying these measures have been proposed in literature. Two such measures are the distribution of transition counts [2, 5, 6, 8] and the gap [7] of a code. Gray codes which are uniform with respect to the former measure are referred to as balanced Gray codes. To make this notion precise, associate with an n bit cyclic Gray code L n = w 1 ; w 2 ; w 2 n , the transition sequence of bit positions s = s 1 ; s 2 ; ....

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V. E. Vickers and J. Silverman, A Technique for Generating Specialized Gray Codes, IEEE Trans. Comput. C-29 (1980) 329-331.

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V.E. Vickers and J. Silverman, A technique for generating specialized Gray codes, IEEE Trans. Comput. C-29(1980) 329--331.

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