J. N. Pierce, "Limit distribution of the minimum distance of random linear codes," IEEE Trans. Inform. Theory, vol. IT--13, pp. 595--599, Oct. 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of codes of the same size. Such arguments have been successfully employed in the past, ever since Shannon employed a random coding argument to prove the channel coding theorem [23] 24, pp. 198 203] That the method can produce quite strong results is to some extent explained by Pierce s results [25], according to which the Gilbert Varshamov bound 7 is tight for almost all binary linear block codes, if n C ( is large. Hence, a random code is a good code. See also [23] and [27] regarding the error probability of random codes. We will now verify the observation in Section V, that G C ( ....

....the left hand side, which measures the proportion of L n R , that does not satisfy G C ( 1 e . o If we constrain our interest to systematic codes only, properties similar to Theorems 7 and 8 can be derived for such a set of codes, too. A useful method to modify the theory was given in [25]. Theorem 8 complements Corollary 6, and together they characterize the curves of Figures 2 and 3. However, as mentioned above, there exist indeed exceptions to the rule of low rate codes having a high neighbor ratio. This will be demonstrated through an example. Consider a code C that is the ....

J. N. Pierce, "Limit distribution of the minimum distance of random linear codes," IEEE Trans. Inform. Theory, vol. IT-13, no. 4, pp. 595--599, Oct. 1967.

....the distance distribution. The notion of binomiality of the distance distribution proves to be useful. It is known, that the distance distribution of random codes is normalized binomial, i.e. the number of code words at distance i from a specific code word equals on average i n i j jCj=2 n [12, 28, 47]. An analysis of the interval where the distance distribution can be upperestimated by the binomial distribution is undertaken in [30, 31] Here we deal with the lower bounds on the distance components. Our estimates improve on results by Kalai and Linial [18] 2.1 A generalization of the ....

J. N. Pierce, "Limit distribution of the minimum distance of random linear codes," IEEE Trans. Inform. Theory, vol.13, 1967, pp.595--599.

....by various researchers, especially for families of random codes with a constant information rate. Such codes can be obtained by randomly filling up a parity check matrix of the appropriate dimension with zeros and ones. It is known that these codes almost surely satisfy the GilbertVarshamov bound ([23]) and therefore that they can correct a constant fraction of the length of the codewords. More accurately, we let ae = k=n and we define a function = GV (ae) by the relation 1 Gamma ae = H 2 ( with 0 1=2 and H 2 (x) Gammax log 2 x Gamma (1 Gamma x) log 2 (1 Gamma x) Then, for any ....

J. N. Pierce. Limit distributions of the minimum distance of random linear codes, IEEE Trans. Inform. Theory, (1967), 595--599.

....of the system. In the present paper we determine bounds on the optimal tradeoff between source and channel coding for classes of channel codes that attain the Gilbert Varshamov bound. It is known that, asymptotically, a random linear code achieves the Gilbert Varshamov bound with probability one [4, 5], although most known structured classes of codes fall short of the bound. The existence of certain Goppa codes, alternant codes, self dual codes, and double circulant or quasi cyclic codes, all of which meet the Gilbert Varshamov bound, has been discussed in [6, p. 557] A significant ....

J. N. Pierce, "Limit Distribution of the Minimum Distance of Random Linear Codes," IEEE Trans. Info. Theory, vol. IT-13, pp. 595--599, October 1967.

....Symmetry Groups, and a [32, 17, 8] Code Ying Cheng Department of Mathematics Louisiana State University Baton Rouge, LA 70803 N. J. A. Sloane Mathematical Sciences Research Center AT T Bell Laboratories Murray Hill, NJ 07974 I. Introduction Since random codes are good ( 1] 27, p. 558] [29]) one wishes to identify families of codes which are large enough to have a chance of including some good codes, yet small enough to be manageable. In this paper we describe one such family: the codes obtained from the action of the automorphism group of the n dimensional cube on its m ....

J. N. Pierce, "Limit distribution of the minimum distance of random linear codes," IEEE Trans. Information Theory, IT-13 (1967), 595-599.

....the antichain condition, no codeword in L may have minimum weight (since ae g 2 ) Thus the number of to remove is decreased (see section 2.3 for estimations of the typical weights of the elements in L) 2 Typical behaviour of g t and d t 2. 1 VG type bounds It is well known (see, e.g. 7] [9]) that asymptotically, almost all codes lie on the Gilbert Varshamov bound, i.e. satisfy R = 1 Gamma H(ffi) where H denotes the binary entropy function. Let us recall one way of proving this so as to make the forthcoming generalization clearer. Choose randomly a code C among all codes with a ....

J.N. Pierce, Limit distribution of the minimum distance of random linear codes, IEEE Transactions on Information Theory 13 (1967) 595-599.

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J. N. Pierce, "Limit distribution of the minimum distance of random linear codes," IEEE Trans. Inform. Theory, vol. IT--13, pp. 595--599, Oct. 1967.

No context found.

J. N. Pierce. Limit distribution of the minimum distance of random linear codes. IEEE Trans. Inform. Theory, 13:595--599, Oct. 1967.

No context found.

J. N. Pierce, "Limit distribution of the minimum distance of random linear codes," IEEE Trans. Inform. Theory, vol. IT-13, pp. 595--599, Oct. 1967.

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