DECEMBER 2019 EXAMINATION DIET
SCHOOL OF MATHEMATICS & STATISTICS
Finite Mathematics代考 You may use, without proof, results from lectures concerning the number of points on each line, provided that they are clearly stated.
MODULE CODE: MT4516
MODULE TITLE: Finite Mathematics
EXAM DURATION: 2 hours
EXAM INSTRUCTIONS: Attempt ALL questions.
The number in square brackets shows the maximum marks obtainable for that question or part-question.
Your answers should contain the full working required to justify your solutions.
PERMITTED MATERIALS: Non-programmable calculator
YOU MUST HAND IN THIS EXAM PAPER AT THE END OF THE EXAM
PLEASE DO NOT TURN OVER THIS EXAM PAPER UNTIL YOU ARE INSTRUCTED TO DO SO.
1.(a) Let C ⊆ Z62be the code with parity check matrix Finite Mathematics代考
(i) Find a generator matrix for C. Hence, or otherwise, determine |C|.
[3]
(ii) Prove that the vectors 100000 and 000101 may be chosen as coset leaders for C.
[3]
(iii) Decode the received word 001100.
[1]
(iv) Determine the error detecting and correcting capabilities of C.
Brieflfly justify your answer. Finite Mathematics代考
[2]
(b) Consider the function E : Z 42→ Z62 given by
E : (x1, x2, x3, x4) 7→ (x1, x2 + x3, x3 + x4, x2 + x3, x3 + x4, x1 + x2).
Show that E is a valid encoding function, and that the image of E is a linear code. (You should assume that every element of Z42 is a message).
[3]
(c) Prove that if a code C ⊆ Zn2 can correct all errors of weight up to k then the minimum distance of C is at most 2k + 1.
[3]
2.(a) Defifine a Latin square, and state what it means for two Latin squares to be orthogonal.
[2]
(b) Find all possible Latin squares whose fifirst three rows are the following Latin rectangle.
[4]
(c) Find a Latin square that is orthogonal to both of the following.
[2]
(d) The following Latin square is a (suitably renumbered) direct product of
two smaller squares.
Find the two smaller squares.
[3]
3.(a) Defifine a projective plane.
[3]
(b) Let (P,L) be a fifinite projective plane, let l ∈ L be a line, and let P be a point such that P∉ l. Prove that the number of points on l is equal to the number of lines through P.
[2]
(c) A quadrangle in a fifinite projective plane is a set of four points such that no three are collinear. Determine the number of quadrangles in a fifinite projective plane of order n 2 + n + 1. You may use, without proof, results from lectures concerning the number of points on each line, provided that they are clearly stated.
[3]
(d) Does there exist a projective plane of order 13? Justify your answer.
[2]
4.(a) Defifine a (v, b, r, k, λ)-design. Finite Mathematics代考
[2]
(b) State the values of (v, b, r, k, λ) for each of the following types of designs.
(i) A fifinite affiffiffine plane of order n2 .
[1]
(ii) A Steiner triple system of order 6n + 1.
[2]
(c) There exists a (6, 10, 5, 3, 2)-design with blocks {1, 2, 3}, {1, 3, 4}, and{1, 4, 5}. Find the remaining blocks.
[4]
(d) Prove that there are exactly two tuples (v, b, r, k, λ) with v = 15, λ = 1,and k < v for which there exists a (v, b, r, k, λ)-design. Fully justify your answer.
[5]
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