   Chapter 2.7, Problem 26E

Chapter
Section
Textbook Problem

Write out the eight code words in the ( 5 , 3 ) code where each code word x 1 x 2 x 3 x 4 x 5 is generated in the following way: x i   ∈   { 0 ,     1 } x 4   ≡   x 1 + x 2 ( mod     2 ) x 5   ≡   x 2 + x 3 ( mod     2 ) .

To determine

The eight code words in the (5,3) code where each code word x1x2x3x4x5 is generated in the following way:

xi{0,1}x4x1+x2(mod2)x5x1+x3(mod2)

Explanation

Given information:

xi{0,1}x4x1+x2(mod2)x5x1+x3(mod2)

Explanation:

Here x1,x2,x3{0,1}

Then x1,x2,x3 has two choices 0 or 1.

There are 23=8 ways to form 3- digit word using {0,1}.

There are 8 cases: 000,001,010,011,100,101,110,111

For x=000, x1=0,x2=0,x3=0

x40+0(mod2)x40(mod2)

x50+0(mod2)x50(mod2)

Therefore, (5,3) code word is 00000.

For x=001, x1=0,x2=0,x3=1

x40+0(mod2)x40(mod2)

x50+1(mod2)x51(mod2)

Therefore, (5,3) code word is 00101.

For x=010, x1=0,x2=1,x3=0

x40+1(mod2)x41(mod2)

x50+0(mod2)x50(mod2)

Therefore, (5,3) code word is 01010.

For x=011, x1=0,x2=1,x3=1

x40+1(mod2)x41(mod2)

x50+1(mod2)x51(mod2)

Therefore, (5,3) code word is 01111

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