please explain step by step please Exercise 1.2=Construct the sets Cn and Coo for the ternary code CDo the same for C = {02, 12, 120, 21}.{02, 12, 120, 20, 21}.

(1) Constructing sets Cn and C∞ for the set C = 02, 12, 120, 20, 21 According to…

Answered: Construct the sets Cn and Coo for the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 13E

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please explain step by step please

Exercise 1.2
=
Construct the sets Cn and Coo for the ternary code C
Do the same for C = {02, 12, 120, 21}.
{02, 12, 120, 20, 21}.

Expert Solution

Step 1

(1) Constructing sets $C_{n}$ and $C_{\infty}$ for the set $C = \{02, 12, 120, 20, 21\}$

According to Sardinas - Patterson Theorem:

$C_{n} = \{w \in T^{+} / u w = v w h e r e u \in C, v \in C_{n - 1} o r u \in C_{n - 1}, v \in C\}$ for

For $C_{n}$ :

$C_{1} = \{0\}$

$C_{2} = \{2\}$

$C_{3} = \{0, 1\}$

$C_{4} = \{2, 20\}$

$C_{5} = \{0, 1\}$

In above code, Repetition of $C_{3} = \{0, 1\}$ = $C_{5} = \{0, 1\}$ occurs.

That gives $C_{n} = \{0, 1\}$ or $C_{n} = \{2, 20\}$ for odd or even ternary $n \geq 3$

For $C_{\infty}$ :

$C_{\infty} = C_{1} \cup C_{2} \cup C_{3} \cup C_{4}$

$\Rightarrow$ $C_{\infty} = \{0, 1, 2, 20\}$

The Sardinas-Patterson Theorem states that C code is uniquely decodable if and only if $C_{n} \cap C_{\infty} \neq \emptyset$ .

Here, $C_{n} \cap C_{\infty} \neq \emptyset$ . So, C code is not uniquely decodable.

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Construct the sets Cn and Coo for the ternary code C = {02, 12, 120, 20, 21}. Do the same for C = {02, 12, 120, 21).