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# Finite Mathematics代考 数学考试助攻

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## 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.

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|.

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(ii) Prove that the vectors 100000 and 000101 may be chosen as coset leaders for C.

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(iii) Decode the received word 001100.

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(iv) Determine the error detecting and correcting capabilities of C.

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(b) Consider the function E : Z 42Z6given 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).

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(c) Prove that if a code C Zncan correct all errors of weight up to k then the minimum distance of C is at most 2k + 1.

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### 2.(a) Defifine a Latin square, and state what it means for two Latin squares to be orthogonal.

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(b) Find all possible Latin squares whose fifirst three rows are the following Latin rectangle.

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(c) Find a Latin square that is orthogonal to both of the following.

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(d) The following Latin square is a (suitably renumbered) direct product of

two smaller squares.

Find the two smaller squares.

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### 3.(a) Defifine a projective plane.

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(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.

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(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.

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(d) Does there exist a projective plane of order 13? Justify your answer.

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### 4.(a) Defifine a (v, b, r, k, λ)-design.  Finite Mathematics代考

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(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 .

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(ii) A Steiner triple system of order 6n + 1.

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(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.

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(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.

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