Mathematics 502
Final (50 points)
Due 5/10/07, 12.00noon
Points (leave blank)
1
2
3
4
5
P
Name:
(clearly, please)
This exam is my own work. Sources (apart from the textbooks and my lecture notes) are
indicated.
Signature
This final is due Thursday, May 10th 2007 at 12.00noon (at my office or in my mailbox).
Notes
• Put your name on this cover sheet and sign it.
• You are permitted to use your notes and any publication (book, journal, web page).
You are not permitted to consult third persons.
Results which are quoted from a publication (apart from the course textbook and your
lecture notes) must be indicated.
• You may use a computer unless this renders the problem trivial.
• If you need extra sheets, staple them to this final. (You do not need to submit scrap
paper.)
• I will put the graded finals with your course grades in your mailboxes in the math
department if you have one. Otherwise you can pick up your exam in the front office
starting the week after finals week.
I can email you final score and grade if you want me to. In this case please write an
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requests for grades if they are not requested here.)
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1) Suppose that D is a t − (v, k, 1) design. Show that the numbers
v
k
v−1
k−1
v−t+1
k−t+1
/
,
/
,...
/
t
t
t−1
t−1
1
1
must be integers. Conclude that if there is a 2 − (v, 3, 1) design we must have that v ≡ 1, 3
(mod 6).
(10 points)
2) You are given 12 coins, one of which is known to be either lighter or heavier than all
the others; you are also given a beam balance. Devise a scheme of three weighings which will
identify the odd coin and determine if it is light or heavy; the coins weighed at each step
should not depend on the results of previous weighings. (What is the connection between
this problem and error-correcting codes over F3 ?)
(10 points)
3) Construct a BCH code over F3 of length 26 and designed distance 5.
(10 points)
2
2
4) For a projective plane of order n we form an incidence matrix A ∈ {0, 1}n +n+1×n +n+1
with rows corresponding to lines and columns to points. Let C be the F2 -code generated by
the rows of this matrix.
a) Show that for odd n this code C consists of all the words of even weight.
b) Now consider n ≡ 2 (mod 4). We form an extended code C 0 by adding to every code
word one parity bit. Show that C 0 ⊂ C 0 ⊥ .
(10 points)
5) A graph X with n vertices is called strongly regular with parameters (n, k, a, b) if it
is regular with vertex valency k, every pair of adjacent vertices has a common neighbors
and every pair of non-adjacent vertices has b common neighbors. (For example the Petersen
graph is strongly regular with parameters (10, 3, 0, 1).) Show that the adjacency matrix A of
a strongly regular graph has at most 3 different eigenvalues.
(10 points)