J. K. Wolf, A. M. Michelson, and A. H. Levesque. On the probability of undetected error for linear block codes. IEEE Transactions on Communications, 30(2), Feb 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....number of codewords in the dual code which are of weight i, then ( LinCostello] Pud (C,epsilon) 2 r Sigma [for i=0 to n] Bi (1 2 epsilon) i (1 epsilon) n Wolf [Wolf94o] introduced an efficient algorithm for enumerating all the codewords of a code and finding their weight distribution. Wolf [Wolf82] found that, counter to what was assumed, 1) there exist codes for which Pud(C,epsilon) Pud(C,0.5) for some epsilon not= 0.5 and (2) Pud is not always increasing for 0 =epsilon =0.5. The value of what was assumed to be the worst Pud is Pud(C,0.5) 2 r) 2 n) This stems from the fact that ....

J.K. Wolf, Arnold Michelson and Allen Levesque. On the probability of undetected error for linear block codes. IEEE Transactions on Communications, COM-30:317-324, 1982

.... Then P (q; n; q k ; ffl) n X w=n Gammak 1 n w (q k w Gamman Gamma 1) ffl q Gamma 1 w 1 Gamma q ffl q Gamma 1 n Gammaw : 4) This theorem improves some previously known bounds on the probability of undetected error and, in particular, a bound for linear codes from [19], 11] A. Proof of Theorem 1. Our observation is that in the linear case this theorem is equivalent to the obvious fact that any submatrix of the parity check matrix of an [n; k] linear code C has rank at most n Gamma k. We begin with reformulating this fact. Let N = f1; 2; ng. For any ....

J. K. Wolf, A.M. Michelson, and A.H. Levesque, " On the probability of undetected error for linear block codes," IEEE Trans. Commun, 30 (1982), 317--324.

.... turns out to be a line segment exiting at Gamma45 o from any point of the bound (12) Hence, if the conjecture on the tightness of the Gilbert Varshamov bound were true, it would imply the tightness of the bounds (10) and (11) For other results on the probability of undetected error see e.g. [5, 19, 20, 21, 22, 23, 24, 25, 34, 35, 54]. About the probability of decoding errors the following is known. The lower bounds are due to Elias, 1956, and Gallager, 1963, 1965, 1968, E de (R; p) n T (ffi GV (R) p) R Gamma 1 if R crit R C p [10] 16) 1 Gamma log i 1 q 4p(1 Gamma p) j Gamma R if R min R R crit [13, 14] ....

J. K. Wolf, A. M. Michelson, and A. H. Levesque, "On the probability of undetected error for linear block codes," IEEE Trans. Commun., vol.30, no.2, 1982, pp.317--324.

.... n Gammaw Gammaj n Gamma w j Bw = n X i=0 x n Gammai y i i X w=1 ( Gamma1) i Gammaw n Gamma w n Gamma i Bw ; where in the last equality we have put i = n Gamma j: Specifying (1) for linear codes, we get Bw = P jEj=w (2 dim C 0 (E) Gamma 1) familiar from [34], 13] Thus, binomial moments of the weight spectrum of C describe the cumulative cardinality of subcodes of C with support of size at most w, see [27] 13] 2] 3] In this form they were implicitly used by MacWilliams [27] in the proof of the celebrated MacWilliams identities. Namely, using ....

.... linear matroids [10] 2] A closely related circle of problems was studied in [31] 11] q analogs of the binomial moments were studied by Delsarte in his work on association schemes of bilinear forms [7] Finally, binomial moments arise in the study of the probability of undetected error, see [34], 1] 3] The potential of the binomial moments that stems from Lemma 1, though made clear by [13] 32] remained largely unclaimed (however, see [14] The distance spectrum coefficients (2) satisfy the Delsarte inequalities [6] n X i=0 P k (i)A i 0; 0 k n; 9) where the numbers P k (i) ....

[Article contains additional citation context not shown here]

J. K. Wolf, A.M. Michelson, and A.H. Levesque, " On the probability of undetected error for linear block codes," IEEE Trans. Commun, 30 (1982), 317--324.

.... Then P (q; n; q k ; ffl) n X w=n Gammak 1 n w (q k w Gamman Gamma 1) ffl q Gamma 1 w 1 Gamma qffl q Gamma 1 n Gammaw : 4) This theorem improves some previously known bounds on the probability of undetected error and, in particular, a bound for linear codes from [19], 11] 2.1. Proof of Theorem 1 Our observation is that in the linear case this theorem is equivalent to the obvious fact that any submatrix of the parity check matrix of an [n; k] linear code C has rank at most n Gamma k. We begin with reformulating this fact. Let N = f1; 2; ng. For ....

....= w X j=1 ( Gamma1) j Gammaw n Gamma j n Gamma w B j ; 0 w n: 7) Clearly, Bw 0. Using (6) rewrite (2) as follows: P (C; ffl) n X w=1 Bw 1 Gamma fflq q Gamma 1 n Gammaw ffl q Gamma 1 w : 8) A second ingredient of the proof is the following Lemma 2 [16] [19], 8] We have Bw = X E2( N w ) jC(E)j Gamma 1) where the summation is over all w subsets of N . Proof: This is a double counting argument. Let E N; jEj = w. Let c 2 C nf0g; wt (c) j; supp (c) E. We can choose such a subset E in Gamma n Gammaj n Gammaw Delta different ways, or ....

J. K. Wolf, A. M. Michelson, and A. H. Levesque, " On the probability of undetected error for linear block codes," IEEE Trans. Commun, 30 (1982) pp. 317--324.

No context found.

J. K. Wolf, A. M. Michelson, and A. H. Levesque. On the probability of undetected error for linear block codes. IEEE Transactions on Communications, 30(2), Feb 1982.

No context found.

J.K. Wolf, Arnold Michelson and Allen Levesque, "On the probability of undetected error for linear block codes", IEEE Transactions on Communications, COM-30: 317-324, 1982.

No context found.

J. K. Wolf, A. M. Michelson, and A. H. Levesque, `On the probability of undetected error for linear block codes', IEEE Trans. Comm.,vol. 30, pp. 317--324, Feb. 1982.

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