Question

Exercise 1 Assume that a binary symmetric channel is used in this question as shown in fig. 1 I-p 0 0 Alice (encoder) Bob (decoder) 1 1-P Figure 1 The Binary symmetric channel where the probability of a bit being wrongly decoded by Bob is p i) Two codes are proposed: a binary 3-fold repetition code i.e. with codewords 000 and 111; and binary 5- fold repetition code i.e. with codewords 00000 and 11111 For each case write down the probability of receiving a word with 1 error in it. Assume that the probability that an error in a randomly chosen digit of a randomly chosen message is 0.035 i) For each of the above codes, is it a one-error correcting code, a two-error correcting code or neither? State any assumptions made.

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Answer 1

art- Ci nas As riven, C e000, 111 Let le behe a wrd with 1 eno ot· 3eceivin hen n n-k 3/

Fer Binang s.hed Rep冶tion As C 00000, 1111 s-1 o 35)-035 。Binang 3-fold Repetition is a 1-erro5 Co ㆅ ectigCode As la f..mu( , n= 3 , Soーニー-1. mpe Sender Sends "111 and eceive s exa Yeceives "llo,, with 1. bit esso., then 1e8Yoǐ can be Cosrectedy talin arfues, while arruminy weaxe Sendin111'bit for 1 and 'ood for it te no of bits as Cossect Bina ej, if we seceivedIlo o' bit we take more no. of its as cossected so hexe we ass ame we Seceive 11., as we丫eceived ζ ‘l's, and a 'o'.

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