coding theory

(b) We can analyse cyclic codes using the fact that they correspond to certain subsets of the ring F,x)/(x" 1). What type of subsets correspond to cyclic codes? What property of those subsets corresponds to the cyclic condition, and why? What characteristic of the code does the parameter n in the ring correspond to? [4 Marks] Suppose is the smallest cyclic code in Z; that con- tains the codeword 1010. List all the codewords in C. [2 Marks (ii) Explain how the ternary Golay code arises. You do not need to give all the coefficients of the irreducible polynomial, but your explanation should specify the length and dimension of the code. 2 Marks (iii) (a) Let D be the elliptic curve 23 + 3x +1 = y2 over 27. List all the points on D. 3 Marks) (b) Find the product of the two points (0,1), (2,6) in the (3 Marks elliptic curve D