Problem set 5
Due date: 05/09/2016
Check carefully which problems you need to solve.
1. Practice problems
P1. Let B be a block design of parameters 2 − (v, k, λ) such that k < v. Show
that the following statements are equivalent:
(i) v = b, where b is the number of blocks.
(ii) r = k, where r is the number of blocks incident to a given point.
(iii) The size of the intersection of any two different blocks is λ.
(iv) The size of the intersection of any two different blocks is the same.
P2. Let the vertices of K16 be labeled by the the subsets of {1, 2, 3, 4, 5} with
even cardinality. Let us define the edge disjoint spanning subgraphs of K16 as
follows:
(i) (S, T ) ∈ E(G1 ) if |(S \ T ) ∪ (T \ S)| = 4.
(ii) (S, T ) ∈ E(G2 ) if (S \ T ) ∪ (T \ S) ⊆ {{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 1}}.
(iii) (S, T ) ∈ E(G3 ) if (S \ T ) ∪ (T \ S) ⊆ {{1, 3}, {2, 4}, {3, 5}, {4, 1}, {5, 2}}.
Show that G1 , G2 and G3 are triangle-free graphs. Show that G1 , G2 and G3
are all strongly regular graphs with the same parameters, and actually they
are isomorphic.
2. Homework problems
1. Construct a Steiner triple system of parameters 2 − (2m − 1, 3, 1) and
2 − (3m , 3, 1). In other words, decompose the complete graph Kn into edge
disjoint triangles if n = 2m − 1 or n = 3m .
2. Construct strongly regular graphs of parameters (16, 5, 0, 2) and (16, 6, 2, 2).
Then from both graphs construct a block design of parameters 2 − (16, 6, 2).
3. Show that for all (n, d, q) there exists a code C with parameters (n, M, d)q
such that
qn
M ≥ Pd−1 n
.
j
(q
−
1)
j=0 j
(Remark: it resembles to the Hamming-bound, but it is not! The end of the
summation is different.)
1
2
4. Let’s play twenty question game! What’s the smallest number of questions
one needs in order to find out a number between 1 and 32 if one has to write
down all questions in advance and you may not get answers to two questions?
All questions must be of the form: "Is the number in the set ...?".
5. Show 6 vertices in the regular 31-gon such that all distances are different
among them.
6. Let H1 , . . . , Hn be subsets of {1, 2, . . . , n} such that |Hi | = k for i =
1, 2, . . . , n, and |Hi ∩ Hj | = λ if i 6= j. Assume that λ < k. Show that every
element of {1, 2, . . . , n} is contained in exactly k sets Hi ’s.
7. Consider the G23 Golay-code with parameters [23, 12, 7]2 . (This is the code
obtained by puncturing G24 at some coordinate.) Let B the set of blocks whose
characteristic vectors are the weight 7 codewords in G23 . What can be the
intersection of B1 , B2 ∈ B? Construct a strongly regular graph of parameters
(253, 140, 87, 65).
8. Let `1 , . . . , `n be equiangular lines in Rd such that any two of them have
angle α. Show that if 1 > d cos2 (α) then
n≤
d(1 − cos2 (α))
.
1 − d cos2 (α)
9. Let C be a code of parameters (7, 16, 3)2 such that 0 ∈ C. Show that these
conditions determine uniquely the code up to permutations of the coordinates.
10. Construct a block design of parameters 2 − (400, 8, 1). In other words,
decompose the complete graph K400 into edge disjoint K8 ’s such that each
edge is exactly in one K8 .
11. Show that the following matrix is the generator matrix of a perfect(!) code
of parameters [11, 6, 5]3 :


1 0 0 0 0 0 1 1 1 1 1
 0 1 0 0 0 0 0 1 -1 -1 1 


 0 0 1 0 0 0 1 0 1 -1 -1 


 0 0 0 1 0 0 -1 1 0 1 -1 
 0 0 0 0 1 0 -1 -1 1 0 1 
0 0 0 0 0 1 1 -1 -1 1 0
3
This is the ternary Golay-code G11 . How can we get a code G12 of parameters
[12, 6, 6]3 from G11 ?
12. (a) Show that it is not possible to choose 11 vertices from the regular
111-gon such that all distances are different among them.
(b) Let n ≡ 1 (mod 3). Show that if there exists n + 1 vertices among the
vertices of a regular (n2 +n+1)-gon such that all distances are different among
them then there exists integers a and b such that n = a2 + ab + b2 .