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- (a) Let e be a positive integer. Define what is meant by an e-perfect code of length n. Suppose that C is an e-perfect linear code of length 9. Find the values of e and dim C. Justify your answer carefully.
(b) Define what is meant by a 2-design with parameters (v, k, λ). Let c be a positive integer. Prove the following statements. (You may use any results from the course provided you state them clearly.) (i) There is no 2-design with parameters (2c^2 − c + 1, 2 c, 1), where c > 1. (ii) If there exists a 2-design with parameters (8c^2 + 6c + 2, 4 c + 2, 2), then c must be a square.
(c) Define what is meant by a strongly regular graph with parameters (v, k, a, b). Prove that there is no such connected graph with the property that v = 2 k + 2. (You may use any results from the course provided you state them clearly.)
- Let C be a linear code of length n.
(a) Prove that if d(C) ≥ 5 then dim C < n + 1 − 2 log 2 n.
(b) Now suppose that d(C) = 4 and that C has check matrix A, a k × n matrix of rank k. (i) Let the columns of A be c 1 ,... , cn. Prove that for any i, j, the sum ci + cj is not a column of A. (ii) Deduce that the number of columns of A is at most 2k−^1. (iii) Hence show dim C ≤ n − log 2 n − 1. (iv) For n = 8, show that equality can hold in part (iii), i.e. there exists a linear code of length 8, minimum distance 4, and dimension 4 (= 8 − log 2 8 − 1).
(You may use any results from the course provided you state them clearly.)
- Let B be a 2-design with parameters (v, k, λ). Let b be the total number of blocks in B, and r the number of blocks containing a given point. Suppose that B is quasisymmetric, i.e. there exist integers x < y such that any two blocks in B intersect in either x or y points (and both x and y occur). Define Γ = Γ(B) to be the graph which has the blocks in B as vertices, two vertices B 1 and B 2 being joined if and only if |B 1 ∩ B 2 | = y. Let M be the v ×b incidence matrix of B, and let A be the b×b adjacency matrix of Γ.
(i) Prove that M M T^ = λJv + (r − λ)Iv, where Jv denotes the v × v matrix in which all entries are 1.
(ii) Prove that M T^ M = (y − x)A + xJb + (k − x)Ib.
(iii) Show that there exist real numbers α, β, γ such that A^2 = αA + βJb + γIb.
Deduce that Γ is strongly regular.
(iv) By multiplying both sides of the equation in (ii) by the column vector (1, 1 ,... , 1)T^ , or otherwise, show that the valency of Γ is
(r − 1)k − (b − 1)x y − x.
