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Homework answers / question archive / 2) An [n, k, d]-linear code C has generator matrix G = G= 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 (a) Construct a three-column encoding table showing the message words, code words and the weights of the code words
2) An [n, k, d]-linear code C has generator matrix G =
G= |
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(a) Construct a three-column encoding table showing the message words, code words and the weights of the code words. [Explain fully how you obtained the message words and the code words.]
(b) What are the parameters n, k and d of the code? [Explain your answer fully.]
(c) How many errors can be detected, and how many corrected? [Justify your answers.)
(d) Let t be the number of correctible errors. For each code word c, construct the t-sphere St(c) of radius t centred at c. That is, list all the elements in each t-sphere. [Explain your answer fully.]
(e) What is the packing density of the code? [Explain your answer fully.]
(F) List all the elements of Zn2 that don't belong to any of the t-spheres.
(g) For each word you've listed in part (f), find its Hamming distances from all the code words.
(h) Can the words in part (f) be corrected? [Explain your answer fully.|
3) A linear code has generator matrix G =
G=
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(a) Find a parity-check matrix H for the code.
(b) Illustrate that your answer has the required properties by applying it to any 5 nonzero codewords and any 5 noncodewords.
4) (a) Construct the finite field of order 16 using a 5-term irreducible quartic polynomial. List the elements of the field in a four-column table, with the fourth column having the vector form of the elements, the third column having the polynomial form in terms of a primitive element α, the second column having the exponential form in terms of α where possible and the first column having the discrete logarithms of the terms in the second column (when they exist). Explain how you have produced the polynomial form from each exponential form where the exponent is greater than 4.
(b) Construct the cyclotomic cosets of 2 modulo 15.
(c) For each cyclotomic coset, multiply together the corresponding linear polynomials over F16 so as to obtain an irreducible polynomial over Z2.
(d) How many different binary polynomials of degree 5 that divide x15 + 1 can be used to construct cyclic codes with minimum distance at least 4?
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