Coding Theory Quiz 1's solutions (2006)

0.04 MB·English

by  KIT Tyabandha

Department of Mathemati
s, Mahidol University Kit Tyabandha, PhD
Quiz 1
Coding Theory
th
20 January 2006
Time: 1 hours (12:30{1:30pm)
1. Write the addition [3℄y and multipli
ation [4℄ tables for Z6.
Solution.The addition table,
+ 0 1 2 3 4 5
0 0 1 2 3 4 5
1 1 2 3 4 5 0
2 2 3 4 5 0 1
3 3 4 5 0 1 2
4 4 5 0 1 2 3
5 5 0 1 2 3 4
#
The multipli
ation table,
(cid:1) 0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4 0 2 4
3 0 3 0 3 0 3
4 0 4 2 0 4 2
5 0 5 4 3 2 1
#
2. Given ISBN 0198538 30. Find the missing digit .[3℄
Solution.ForISBN x1:::x10,
X10
ixi (cid:17)(mod11)
i=1
Writing y for ,
0+1(2)+9(3)+8(4)+5(5)+3(6)+8(7)+y(8)+3(9)=187+8y (cid:17)0(mod11)
Hen
e y =0, and the ISBN is therefore 0198538030.
#
2 3
3. Let f(x)=1+x +x . Showwhetherf(x) isirredu
ible overZ2.[4℄Then (cid:12)nd Z2[x℄=(f(x)).[4℄
And then draw the addition [5℄ and multipli
ation [7℄ tables of Z2[x℄=(f(x)).
Solution.Wenotethatf(x)isofdegree3. Supposef(x)beredu
ible. Thenitwouldhavealinear
fa
tor x or 1+x, whi
h would make 0 and 1 roots of f(x). But g(0) = g(1) = 1, whi
h is in Z2.
Therefore f(x) is irredu
ible.
#
(cid:0) (cid:1) (cid:8) (cid:9)
2 3 2 2 2 2
Z2[x℄= 1+x +x = 0;1;x;1+x;x ;x+x ;1+x ;1+x+x
#
y Numbers between square bra
kets are marks.
th
Coding theory, Quiz 1 {1{ From 19 Jan 05, as of 20 January, 2006
Department of Mathemati
s, Mahidol University Kit Tyabandha, PhD
The addition table,
2 2 2 2
+ 0 1 x 1+x x x+x 1+x 1+x+x
2 2 2 2
0 0 1 x 1+x x x+x 1+x 1+x+x
2 2 2 2
1 1 0 1+x x 1+x 1+x+x x x+x
2 2 2 2
x x 1+x 0 1 x+x x 1+x+x 1+x
2 2 2 2
1+x 1+x x 1 0 1+x+x 1+x x+x x
2 2 2 2 2
x x 1+x x+x 1+x+x 0 x 1 1+x
2 2 2 2 2
x+x x+x 1+x+x x 1+x x 0 1+x 1
2 2 2 2 2
1+x 1+x x 1+x+x x+x 1 1+x 0 x
2 2 2 2 2
1+x+x 1+x+x x+x 1+x x 1+x 1 x 0
#
The multipli
ation table,
2 2 2 2
(cid:1) 0 1 x 1+x x x+x 1+x 1+x+x
0 0 0 0 0 0 0 0 0
2 2 2 2
1 0 1 x 1+x x x+x 1+x 1+x+x
2 2 2 2
x 0 x x x+x 1+x 1 1+x+x 1+x
2 2 2 2
1+x 0 1+x x+x 1+x 1 1+x+x x x
2 2 2 2 2
x 0 x 1+x 1 1+x+x x 1+x x+x
2 2 2 2 2
x+x 0 x+x 1 1+x+x x 1+x x 1+x
2 2 2 2 2
1+x 0 1+x 1+x+x x 1+x x x+x 1
2 2 2 2 2
1+x+x 0 1+x+x 1+x x x+x 1+x 1 1+x
#
th
Coding theory, Quiz 1 {2{ From 19 Jan 05, as of 20 January, 2006