F. Gray. Pulse Code Communication. March 17, 1953. USA Patent 2,632,058.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we mention a more familiar partial order on the same set: the inclusion relation. This de nes a poset on the subsets of [n] called the Boolean lattice B(n) Its cover graph is the n cube. The n cube is Hamiltonian and has Hamiltonian paths satisfying a wide variety of constraints (e.g. [2, 3, 4, 5, 7]) It is easy to obtain a Hamiltonian path in B(n) by induction. The cover graph of M(n) has a somewhat more complicated structure than that of B(n) and fewer edges: n 1)2 instead of n2 . 2 Necessary Conditions Let A n be the cover graph of M(n) For compactness, we will represent the ....

F. Gray, Pulse code communication, U.S Patent No. 2632058.

....searches for trellis codes pick a single constellation labeling and search over a set of convolutional codes. While the choice of convolutional code is the result of an exhaustive search, the choice of labeling is often not justified beyond stating that the labeling is a Gray code (GC) labeling [1] or a set partitioning (SP) labeling [2] 3] A GC labeled constellation is one where any two points that are nearest neighbors have binary labels that differ in exactly one bit position. The SP labeling paradigm partitions the constellation into a collection of mutually exclusive, collectively ....

....composite difference list in the construction combines two identical difference lists. The difference lists for SU , and PAM below illustrate the recursive structure of SU difference lists. 2) 3) 4) 5) The commonly used reflected binary labeling structure in Gray s 1953 patent [1] is SU with unit Hamming weight difference labels . However, for the general SU structure, can be any basis of linearly independent bit difference labels. Looking back at Fig. 3(a) and (b) both are ultracomposite, but only Fig. 3(a) is SU. Furthermore, Fig. 4(a) is SU, but Fig. 4(b) is not even ....

F. Gray, "Pulse code communications," U.S. Patent 2 632 058, Mar. 1953.

....d(w 2M ,n) d(v 2M ,n) for n =0, 2M iv) d(u 2M ,n) d(bM ,nmod(M) for n =0, 2M A. Binary Reflected Gray Mapping A particular mapping of bits onto symbols that has the appealing property that adjacent bit vectors are separated by one single bit was patented in 1953 by Frank Gray [13]. A great wealth of work has been done to study the general class of Gray mappings (or Gray codes) 14] However, the original scheme proposed by Gray is almost ubiquitous in communications. This particular mapping is referred to as the binary reflected Gray code (BRGC) the name stemming from the ....

F. Gray, "Pulse code communications," US Patent No. 2632058.

....to the Game Of Hanoi. We give an example where these results were put to good use, and indicate when in general the method described can be applied. 1 Introduction Given the fact that similar bitstrings yield widely di#erent decimal numbers led Frank Gray to define and patent the Gray Code [2]: he devised a coding of decimal numbers into bitstrings such that two decimal numbers k and k 1arecodedby bitstrings that di#er in exactly one position. This code was used to reduce the importance of transmission errors. In this paper we generalize this method to codings of decimal numbers by ....

F. Gray. Pulse code communication, Mar. 17 1953. U.S. patent no. 2,632,058.

....A number of schemes exist to code the value that is to be sent in a form that consumes less energy than the traditional twos complement representation. In a signed magnitude representation, the value is separated into an unsigned magnitude portion and a 1 bit sign. The Gray code scheme [4] takes advantage of the fact that values in a sequence tend to be near. Gray codes of consecutive values differ by only one bit. This makes them especially useful for address busses. 1 Anil Bahuman is affiliated with the Artificial Intelligence center at UGA In Bus invert coding [3] ....

F. Gray "Pulse Code Communication", U.S. Patent 2,632,058, March. 1953.

....study how a class of functions using the fundamental operators can be implemented efficiently under this encoding. 37 The Gray coding was invented by Dr. Frank Gray to lower the data loss when transmitting signals across noisy wires. The coding was patented by his employer, Bell Labs, in 1953 [Gra53] The reflected Gray coding of a PowerList permutes the elements in such a way that neighboring elements in the original PowerList are placed in positions of the coded PowerList whose indices written as a binary string only differ in one position. We define the permutation function gray : ....

Frank Gray. Pulse code communication. U.S. Patent 2,632,058, 1953.

....mutated via random bit flips. When such a mutation operator acts upon conventional binary numbers, mutated phenotypic traits are poorly correlated with parental phenotypic traits, since a single bit flip may result in a large change in the value for which the bit string codes. Gray coding [7], through ensuring that consecutive integers are coded for by adjacent binary strings, increases the correlation between mutants and their parents. In addition to sharing the general appeal of context sensitive mutation operators, this coding scheme is advocated by genetic algorithm designers, who ....

Gray, F.: Pulse code communication. U.S. Patent 2 632 058 (1953)

.... a i Delta Delta Delta a 0 , which denotes the string w with the i th bit changed. One Hamiltonian path in a Hypercube is known as the (binary reflected) Gray code. This sequence of nodes (bit string labels) is named after Frank Gray who used it in the design of an A D conversion circuit [5]. However, this sequence was recognized in the last century as a solution method to an old puzzle known as the Chinese Rings; see [2, 3] for discussions of this connection. The Gray code sequence can be recursively described using a method we call recursive reversing. Recursive reversing is a ....

F. Gray, Pulse code communication, 1953, U.S. Patent Number 2,632,058.

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F. Gray, Pulse Code Communication, U. S. Patent No. 263

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F. Gray, Pulse code communication, U. S. Patent 2,632,058, 1958.

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F. Gray, Pulse Code Communication, U. S. Patent No. 263

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F. Gray, "Pulse code communications," U. S. Patent No. 2632058.

....n on the alphabet f0; 1g; two vertices are adjacent if they di er in exactly one coordinate. The transition of an edge vw in Q n is the index vw 2 f1; 2; ng of the coordinate (or bit) in which v and w di er. An n bit (cyclic, binary) Gray code is a Hamilton circuit in Q n . Frank Gray [3] described an elementary family of re ected Gray codes (RGC) which has seen countless applications. Certain applications in engineering, statistics and computer science require specialized Gray codes with properties not possessed by the RGC. We refer to Savage [6] for more information on such ....

F. Gray, Pulse Code Communication, U. S. Patent No. 263

....the codewords all adjacent codewords differ in exactly one bit position, including the first and the last codeword in the list. This property is desirable for M PSK, where we want to label the quantization of a circle. The BRGC scheme, originally proposed and patented by Frank Gray in 1953 [7], is actually only one special code in a large class of codes having the property that adjacent codewords differ in only one position [8, 9] For m =1(M =2) the BRGC is simply 0, 1 . The BRGC of order m can be constructed recursively from the BRGC of order m 1 according to the following ....

F. Gray, "Pulse code communications," U. S. Patent No. 2632058.

....Gray code, Hamiltonian cycle, n cube 1 Introduction A binary Gray code is a listing of all the n bit binary strings, for a given n, such that only one bit changes. between successive items in the list, including the rst and last. The classical approach, known as the binary re ected Gray code [Gil58, Gra58], starts with the 1 bit Gray code 0; 1 ; for n 1, construct the n bit Gray code by rst appending 0 to each element of the (n 1) bit Gray code, then list the (n 1) bit Gray code in reverse, appending 1 to each element. It is easy to see how this produces a complete listing of all n bit strings, ....

F. Gray. Pulse code communications. U.S. Patent 2632058, March 1958.

....j ( G Rf(S j ( unless j 1 = j 1 ; similarly, the edge fn 1; j 1 g does not appear in G S N (Rf(S j ( unless j 1 = j 1 . 3) This follows immediately from (2) Example. Re ecting (0; 1) n 1 times yields the standard re ected Gray code Rn (as featured in F. Gray s patent [5]) Repeatedly applying Proposition 2.1 gives GRn = GRn = K 1;n 1 (a star graph with one central vertex connected to n 1 leaves) as noted in [1, 9] These codes are sucient. Remark. Whenever is sucient, j 1 6= j 1 , and k 6 0; N (mod 2N ) the graph induced by S k (Rf(S j ( ....

F. Gray, Pulse code communication, U. S. Patent 2,632,058, 1958.

....from statement (2) GRAPHS INDUCED BY GRAY CODES 7 Example. Reflecting the 1 bit Gray code (0; 1) n Gamma 1 times yields the standard reflected n bit Gray code Rn , defined for every n 1. These codes, which have many remarkable properties, are those featured in F. Gray s 1953 patent [6]. Because (Rn ) Rf n Gamma1 ( 1) Rn ) must start and end with 1. Proposition 3.1 now implies that GRn = GRn = K 1;n Gamma1 ; a star with n Gamma 1 leaves, as previously noted [12, 1] Example. The code P 4 shown in Figure 1 has transition sequence Rf(S 1 (R 3 ) Rf( 2; 1; 3; 1; ....

Gray, F. Pulse code communication. U. S. Patent 2,632,058. 1958.

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F. Gray. Pulse Code Communication. March 17, 1953. USA Patent 2,632,058.

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F. Gray, "Pulse Code Communications," U.S. Patent 2632058.

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