Y.S.Han Cyclic codes A (7,4)Cyclic Code Generated by g(x)=1+x+23 Messages Code Vectors Code Polynomials (0000) 00000000=0·g(x) (1000) 11010001+x+x3=1·g(x) (0100) 0110100x+x2+x4=x·g(x) (1100) 1011100 1+x2+x3+x4=(1+x)·g(x) (0010) 0011010 x2+x3+x5=x2·g(c) (1010) 1110010 1+x+x2+x5=(1+x2)·9(x) (0110) 0101110 x+x3+x4+x5=(x+x2)·g(x) (1110) 1000110 1+x4+x5=(1+x+x2)·g(x) (0001) 0001101 x3+x4+x6=x3·g(c) (1001) 11001011+x+x4+x6=(1+x3)·g(x) (0101) 0111001 x+x2+x3+x6=(x+x3)·g(x) (1101) 1010001 1+x2+x6=(1+x+x3)·g(x) (0011) 0010111x2+x4+x5+x6=(x2+x3)·g(x) (1011) 1111111 1+x+x2+x3+x4+x5+x6=(1+x2+x3)·g(x) (1111) 1001011 1+x3+x5+x6=(1+x+x2+x3)·g(c) School of Electrical Engineering Intelligentization,Dongguan University of Technology Y. S. Han Cyclic codes 4 A (7, 4) Cyclic Code Generated by g(x) = 1 + x + x 3 Messages Code Vectors Code Polynomials (0 0 0 0) 0 0 0 0 0 0 0 0 = 0 · g(x) (1 0 0 0) 1 1 0 1 0 0 0 1 + x + x 3 = 1 · g(x) (0 1 0 0) 0 1 1 0 1 0 0 x + x 2 + x 4 = x · g(x) (1 1 0 0) 1 0 1 1 1 0 0 1 + x 2 + x 3 + x 4 = (1 + x) · g(x) (0 0 1 0) 0 0 1 1 0 1 0 x 2 + x 3 + x 5 = x 2 · g(x) (1 0 1 0) 1 1 1 0 0 1 0 1 + x + x 2 + x 5 = (1 + x 2 ) · g(x) (0 1 1 0) 0 1 0 1 1 1 0 x + x 3 + x 4 + x 5 = (x + x 2 ) · g(x) (1 1 1 0) 1 0 0 0 1 1 0 1 + x 4 + x 5 = (1 + x + x 2 ) · g(x) (0 0 0 1) 0 0 0 1 1 0 1 x 3 + x 4 + x 6 = x 3 · g(x) (1 0 0 1) 1 1 0 0 1 0 1 1 + x + x 4 + x 6 = (1 + x 3 ) · g(x) (0 1 0 1) 0 1 1 1 0 0 1 x + x 2 + x 3 + x 6 = (x + x 3 ) · g(x) (1 1 0 1) 1 0 1 0 0 0 1 1 + x 2 + x 6 = (1 + x + x 3 ) · g(x) (0 0 1 1) 0 0 1 0 1 1 1 x 2 + x 4 + x 5 + x 6 = (x 2 + x 3 ) · g(x) (1 0 1 1) 1 1 1 1 1 1 1 1 + x + x 2 + x 3 + x 4 + x 5 + x 6 = (1 + x 2 + x 3 ) · g(x) (1 1 1 1) 1 0 0 1 0 1 1 1 + x 3 + x 5 + x 6 = (1 + x + x 2 + x 3 ) · g(x) School of Electrical Engineering & Intelligentization, Dongguan University of Technology