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(b) Use (a) to show that W₂(X) = 1 ((1 ((1 + X)Wc(X) + (1 − X)Wc(−X)).

(b) Use (a) to show that W₂(X) = 1 ((1 ((1 + X)Wc(X) + (1 − X)Wc(−X)).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Task (b) only please. Thanks

Let C be a binary linear code of length n, n odd, and let Ĉ be the extended binary linear code obtained by adding a parity-check symbol to the codewords in C. Let be the weight enumerator of C (recall that A; is the number of codewords of weight i in C). Let be the weight enumerator of Ĉ. (a) Show that n Wc(X) = Σ A₁Xi i=0 = n+1 W₂(X) = Σ Â₁Xi i=0 Â。 = 1; Â; = 0 if i is odd and 1 ≤ i ≤ n; ¡ = A₁-1 + A; if i is even and 2 ≤ i ≤ n -1; • Anti An.
Transcribed Image Text:Let C be a binary linear code of length n, n odd, and let Ĉ be the extended binary linear code obtained by adding a parity-check symbol to the codewords in C. Let be the weight enumerator of C (recall that A; is the number of codewords of weight i in C). Let be the weight enumerator of Ĉ. (a) Show that n Wc(X) = Σ A₁Xi i=0 = n+1 W₂(X) = Σ Â₁Xi i=0 Â。 = 1; Â; = 0 if i is odd and 1 ≤ i ≤ n; ¡ = A₁-1 + A; if i is even and 2 ≤ i ≤ n -1; • Anti An.
(b) Use (a) to show that Wa(X) 1)) { - = ((1 + X)Wc(X) + (1 − X)Wc(−X)).
Transcribed Image Text:(b) Use (a) to show that Wa(X) 1)) { - = ((1 + X)Wc(X) + (1 − X)Wc(−X)).
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