Question A [10 marks in total]. Louise needs to send to Madhu sequences which consist of lettersT, E, B, L. Each letter is represented by a two-bit message, as follows:T= [0 0], E= [0 1], B= [1 0], L= [1 1].Since Louise and Madhu have to use a one-way, noisy channel, BSC(𝑝) with small 𝑝, Louise encodeseach message using the generator matrix𝐺 = [1 0 0 1 10 1 1 0 1]of a binary linear code 𝐶.A1 [2 marks] Represent the word LET as three two-bit messages, encode each message as describedabove and write down the resulting three codevectors of 𝐶.A2 [2 marks] Write down a standard array for 𝐶.
Question
Question A [10 marks in total]. Louise needs to send to Madhu sequences which consist of lettersT, E, B, L. Each letter is represented by a two-bit message, as follows:T= [0 0], E= [0 1], B= [1 0], L= [1 1].Since Louise and Madhu have to use a one-way, noisy channel, BSC(𝑝) with small 𝑝, Louise encodeseach message using the generator matrix𝐺 = [1 0 0 1 10 1 1 0 1]of a binary linear code 𝐶.A1 [2 marks] Represent the word LET as three two-bit messages, encode each message as describedabove and write down the resulting three codevectors of 𝐶.A2 [2 marks] Write down a standard array for 𝐶.
Solution
A1:
First, we need to represent the word "LET" as three two-bit messages.
L = [1 1] E = [0 1] T = [0 0]
Next, we encode each message using the generator matrix G.
For L = [1 1], we multiply it with G to get the codevector.
[1 1] * G = [1 1 1 0 1]
For E = [0 1], we multiply it with G to get the codevector.
[0 1] * G = [0 1 1 0 1]
For T = [0 0], we multiply it with G to get the codevector.
[0 0] * G = [0 0 0 0 0]
So, the resulting three codevectors of C are [1 1 1 0 1], [0 1 1 0 1], and [0 0 0 0 0].
A2:
A standard array for C is a table that lists all the possible codevectors. Since we are dealing with a binary linear code, the standard array will have 2^n rows, where n is the length of the codevectors.
In this case, the codevectors are of length 5, so the standard array will have 2^5 = 32 rows.
The first row is always the zero vector, and the remaining rows are the non-zero codevectors in any order.
Here is an example of what the standard array might look like:
0 0 0 0 0 1 1 1 0 1 0
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