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Math 327 Assignment 5

Marh 327, Assignment 5 about latin squares and BIDB's
Academic year: 2021/2022
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Technische Universiteit Delft

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Math 327 Assignment 5 Due Dec. 1, 2021 Total: 50 marks.

1 Latin Squares

  1. In this question you will construct a finite field of order 8, starting with the fieldZ 2. All calculation should in be Z 2 (use the addition and multiplication table forZ 2 ). (a) (3 points) Show thatx 3 +x+ 1cannot be factored in a nontrivial way inZ 2 (so it cannot be factored into polynomials with coefficients inZ 2 ). (b) (1 point) Letjbe the root of the polynomialx 3 +x+1adjoined toZ 2 (sojis defined so thatj 3 +j+1 = 0). Use this polynomial to construct a field with 23 = 8elements. List the 8 elements in the field. (c) (6 points) Do the following calculations in the field (your solutions should be one of the 8 elements in the field from the previous part). 1.(1 +j) + (1 +j+j 2 ) 2.(1 +j 2 ) + (1 +j 2 ) 3− 1 4 2 (1 +j+j 2 ) 5.(1 +j)(1 +j+j 2 ) 6.(1 +j)− 1

  2. (5 points) Construct three mutually orthogonal Latin squares of order 7.

  3. LetAbe a Latin square of ordernfor which there exists a Latin squareBof ordernsuch thatAandBare orthogonal. ThenBis called anorthogonal mateofA. (a) (1 point) Write out the addition table forZ 4. (Note that this is a Latin square.) (b) (4 points) Prove that the addition table ofZ 4 does not have an orthogonal mate.

  4. (5 points) LetMbe anm×mmatrix andNann×nmatrix. The matrixM⊗Nis defined to be amn×mn matrix. If the(i, j)-entry ofMismi,j, then it is replaced withmi,jNinM⊗N. For example, if

M=

(

2 3

5 7

)

N=

(

a b c d

)

Then

M⊗N=

(

2 N 3 N

5 N 7 N

)

=

2 a 2 b 3 a 3 b 2 c 2 d 3 c 3 d 5 a 5 b 7 a 7 b 5 c 5 d 7 c 7 d

LetA 1 andA 2 be MOLS of ordermand letB 1 andB 2 be MOLS of orderm. Prove thatA 1 ⊗B 1 andA 2 ⊗B 2 are MOLS of ordermn.

  1. (5 points) Does there exist a BIBD with parametersb= 10,v= 8,r= 5, andk= 4? Give a reason why. Does there exist a BIBD whose parameters satisfyb= 20,v= 18,k= 9, andr= 10? Give a reason why.

  2. LetBbe a BIBD with parametersb, v, k, r, λwhose set of varieties areX={x 1 , x 2 ,... , xv}and whose blocks areB 1 , B 2 ,... , Bb. For each blockBi, letBci, denote the set of varieties which do not belong toBi. LetBcbe the collection of subsets{B 1 c, B 2 c,... , Bbc}ofX. The BIBDBcis called the complementary design ofB. (a) (3 points) Determine the complementary design of the Fano plane.

(b) (5 points) Prove thatBcis a block design with parameters

b′=b, v′=v, k′=v−k, r′=b−r, λ=b− 2 r+λ,

provided thatb− 2 r+λ > 0. (c) (2 points) How are the incidence matrices of a BIBD and its complement related?

  1. (5 points) Show thatB={ 0 , 1 , 3 , 9 }is a difference set inZ 13 , and use this difference set as a starter block to construct an SBIBD. Identify the parameters of the block design.

  2. (5 points) In class we saw that if there are Steiner triple systems of indexλ= 1withvandwvarieties, respec- tively, then there is a Steiner triple system of indexλwithvwvarieties. Use this theorem to construct a Steiner triple system of index 1 having 21 varieties.

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Math 327 Assignment 5

Course: Introductory Combinatorics (MATH 327)

University: University of Regina

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Math 327 Assignment 5
Due Dec. 1, 2021
Total: 50 marks.
1 Latin Squares
1. In this question you will construct a finite field of order 8, starting with the field Z2. All calculation should in be
Z2(use the addition and multiplication table for Z2).
(a) (3 points) Show that x3+x+ 1 cannot be factored in a nontrivial way in Z2(so it cannot be factored into
polynomials with coefficients in Z2).
(b) (1 point) Let jbe the root of the polynomial x3+x+1 adjoined to Z2(so jis defined so that j3+j+1 = 0).
Use this polynomial to construct a field with 23= 8 elements. List the 8elements in the field.
(c) (6 points) Do the following calculations in the field (your solutions should be one of the 8 elements in the
field from the previous part).
1. (1 + j) + (1 + j+j2)
2. (1 + j2) + (1 + j2)
3. j−1
4. j2(1 + j+j2)
5. (1 + j)(1 + j+j2)
6. (1 + j)−1
2. (5 points) Construct three mutually orthogonal Latin squares of order 7.
3. Let Abe a Latin square of order nfor which there exists a Latin square Bof order nsuch that Aand Bare
orthogonal. Then Bis called an orthogonal mate of A.
(a) (1 point) Write out the addition table for Z4. (Note that this is a Latin square.)
(b) (4 points) Prove that the addition table of Z4does not have an orthogonal mate.
4. (5 points) Let Mbe an m×mmatrix and Nan n×nmatrix. The matrix M⊗Nis defined to be a mn ×mn
matrix. If the (i, j)-entry of Mis mi,j , then it is replaced with mi,j Nin M⊗N. For example, if
M=2 3
5 7N=a b
c d
Then
M⊗N=2N3N
5N7N=
2a2b3a3b
2c2d3c3d
5a5b7a7b
5c5d7c7d
Let A1and A2be MOLS of order mand let B1and B2be MOLS of order m. Prove that A1⊗B1and A2⊗B2
are MOLS of order mn.
5. (5 points) Does there exist a BIBD with parameters b= 10,v= 8,r= 5, and k= 4? Give a reason why. Does
there exist a BIBD whose parameters satisfy b= 20,v= 18,k= 9, and r= 10? Give a reason why.
6. Let Bbe a BIBD with parameters b, v, k, r, λ whose set of varieties are X={x1, x2, . . . , xv}and whose blocks
are B1, B2, . . . , Bb. For each block Bi, let Bc
i, denote the set of varieties which do not belong to Bi. Let Bcbe
the collection of subsets {Bc
1, Bc
2, . . . , Bc
b}of X. The BIBD Bcis called the complementary design of B.
(a) (3 points) Determine the complementary design of the Fano plane.

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