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M3P17 Exam 2007: Coding Theory, Designs, and Strongly Regular Graphs, Exams of Mathematics

University of AllahabadMathematics

The m3p17 exam from 2007, which covers various topics in coding theory, designs, and strongly regular graphs. It includes questions on linear codes, perfect codes, t-designs, graph theory, and 2-dimensional subspaces of vector spaces.

Typology: Exams

2012/2013

Uploaded on 02/23/2013

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M3P17 exam 2007
1. Define what is meant by a linear code of length n, dimension k, which
corrects eerrors.
State (but do not prove) the Hamming bound and the Gilbert-Varshamov
bound.
For each of the following statements, say whether it is true or false, justifying
your answer. State any standard results you use.
(i) There exists a linear code of length 20 and dimension 13 which corrects 2
errors.
(ii) There exists a linear code of length 20 and dimension 9 which corrects 2
errors.
(iii) There exists a linear code of length 10 and dimension 3 which corrects
2 errors.
(iv) There exists a linear code of length 10 and dimension 2 which corrects
3 errors.
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M3P17 exam 2007

  1. Define what is meant by a linear code of length n, dimension k, which corrects e errors. State (but do not prove) the Hamming bound and the Gilbert-Varshamov bound. For each of the following statements, say whether it is true or false, justifying your answer. State any standard results you use. (i) There exists a linear code of length 20 and dimension 13 which corrects 2 errors. (ii) There exists a linear code of length 20 and dimension 9 which corrects 2 errors. (iii) There exists a linear code of length 10 and dimension 3 which corrects 2 errors. (iv) There exists a linear code of length 10 and dimension 2 which corrects 3 errors.
  1. Define the following: an e-perfect code of length n a t-design. Suppose that C is a linear code of length 24, dimension 12, and minimum distance 8. (a) Let C′^ be the code of length 23 consiting of all the codewords in C with their last digit deleted. Prove that C′^ is 3-perfect. (b) Let X be the set of 24 coordinate positions in Z^242 , and for a codeword c ∈ C of weight 8, define Bc to be the subset of X consisting of the positions of the eight 1’s in c. Define B to be the collection of all such subsets Bc (c ∈ C). Prove that B is a 5-design. (c) Deduce that the number of codewords in C of weight 8 is equal to 759.
  1. Define what is meant by a strongly regular graph with parameters (v, k, a, b). Let Γ be a strongly regular graph with parameters (v, k, 0 , 3), and assume k > 3. Stating any standard results you require, prove the following. (i) v = 13 (k^2 + 2k + 3). (ii) 4k − 3 is a square (of an integer). (iii) √ 4 k − 3 divides k^2. (iv) k = 21.
  1. Let V = Zn 2 , a vector space of dimension n over Z 2. Assume that n ≥ 3. (i) State and prove a formula for the number of 2-dimensional subspaces of V. (ii) Define a design as follows: the points are the vectors in V , and the blocks are all subsets of the form v + W , where v ∈ V and W is a 2-dimensional subspace of V. (Recall that v + W = {v + w : w ∈ W }.) Prove that these blocks form a 3-design, and find its parameters. (iii) Calculate the total number of blocks of the design in part (ii). Given a pair of vectors v, w ∈ V , how many blocks are there containing both v and w? (iv) Prove that the design in part (ii) is not a 4-design.