Mathematics Coding Theory

show all working and theories used to obtain answer

Question 1.

Let C be the linear code generated by 11111, 10011, 01100, 10001. For each of

C and C ? : Calculate generating and parity check matrices, and determine (n, k, d ). Question 2. 110010. (a) Calculate generating and parity check matrices for C . (b) C is an (n, k, d )-code. Determine n, k and d . What is the rate of C ? (c) How many errors can C detect? How many can it correct? (d) Calculate the cosets of C , their syndromes and coset leaders and give the SDA for C . (e) Words w1 = 111010 and w2 = 001101 are received. Using the SDA, determine the code words sent in each case, if possible, or explain why unique decoding is not possible. (f) Determine the original message word for each of w1 , w2 , if possible. Question 3.

n be non-empty subsets. If S ? T , show that T ? ? S ? . (a) Let S , T ? F2 n be a non-empty subset. Show that S (b) Let S ? F2 ?

Let C be the linear code generated by 101011, 001110, 111100, 100101, and

= S ? . Hint: a typical element of

(c) (d)

i ai si with ai ? F2 and si ? S . n is a subspace then V ? (V ? )? . Show that if V ? F2 In (c) put W = V ? and use the fact that dim W + dim W ?

S has the form

= n = dim V + dim V ? to

prove V = (V ? )? .