E. N. Gilbert. A comparison of signaling alphabets. AT&T Technical Journal, 31:504--522, 1952.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....most famous mo of which is probably the Gilbert channel [2] proposed in 1960. Those channel mod)S d)SIH8 ed with Markov chains arecalled channels with memory(includ; channels with zero memory, i.e. memoryless ones . This work s achievement enables us toextend Gilbert s co performance evaluation [3] in 1952, alandE5) thato#ered the well known Gilbert bound dnd4#)z its relationship to the (memoryless binary symmetric channel,and has been serving as aguid for the Hamming metric based de Ha of error correcting cod)S to the case of the burst metric based co d) burst errorcorrecting co d) and ....

....lomesp Shannon s proof of the channel coding theorem is based on an ingenious technique wel#= p wn as random coding, supplpzI noexplNzp construction of good codes. On the binary symmetric channel giving upto make the information rate approach very near the channel capacity 1 h(p) Gil ert [3] provided anexplI=I construction that made the probabilN y of decoding error go to 0, though the Manuscript received January 22, 1999. Manuscript revised April 19, 1999. TheautB iswit tt Graduat School ofInformat]B Systfor ts Universit y ofElectUgB0WU unicatgB0W Chofushi, 182 8585 Japan. # ....

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E.N. Gilbert, "A comparison of signaling alphabets," The Bell System Technical Journal, vol.31, pp.504--522, May 1952.

....ffi 1 let R(ffi) lim sup n 1 R(n; d n ) where d n = ffin(1 o(1) Here and elsewhere in the paper all o(1) terms are taken for n 1. As usual, the entropy function is H(x) Gammax log x Gamma (1 Gamma x) log(1 Gamma x) The best known lower bound for R(ffi) goes back to Gilbert [8] R(ffi) 1 Gamma H(ffi) 1) Gilbert s proof of this bound is simply to grow a code, by always adding new code words subject only to the constraint that no distances smaller than ffin occur. Despite its extreme simplicity, this argument has never been improved, and some researchers believe that ....

E. N. Gilbert, A comparison of signaling alphabet, Bell Syst. Tech. J. 31(1952), 504-522.

.... the same distance from the origin) 36 2 2 2 p 3 Figure 4 2: The cubic packing p 2 p 2 2 Figure 4 3: The octahedral packing 37 If we don t ask for the points to belong to a lattice with minimum distance , an exponential lower bounds for any =ae p 2 is already implicit in Gilbert bound [28] for binary codes. Non constructive proofs for spherical codes were given by Shannon [75] and Wyner [80] However, the points generated by these constructions do not form a lattice. We give a proof of lower bound 3 in which the points are vertices of the fundamental parallelepiped of a lattice ....

E. N. Gilbert. A comparison of signaling alphabets. AT&T Technical Journal, 31:504--522, 1952.

.... problem of finding a large set C ae f0; 1g n such that for any two strings s 1 ; s 2 2 C, the Hamming distance d(s 1 ; s 2 ) i.e. the number of bits where s 1 differs from s 2 ) is smaller than 8n(ffl Gamma 1) Such a set C is indeed an error correcting code and a classical result of Gilbert [3, 6] allows us to state that, for any ffl 17=16, an error correcting code C exists with log jCj (1 Gamma H(8(ffl Gamma 1) n=8 where H is the entropy function defined as H(x) Gammax log x Gamma (1 Gamma x) log(1 Gamma x) for 0 x 1=2. We have thus shown the following result Theorem ....

Gilbert, E.N. "A comparison of signaling alphabets". Bell System Tech. J., 31:504-- 522, 1952.

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E. N. Gilbert. A comparison of signaling alphabets. AT&T Technical Journal, 31:504--522, 1952.

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