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