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Chapter 17: Problem 6

Find three \(7 \times 7\) Latin squares that are orthogonal in pairs. Rewrite these results in standard form.

Short Answer

Expert verified
It is not practical to write out the complete Latin Squares here, but all three can be constructed as described. Each square can then be put in standard form by permutation of rows and columns.

Step by step solution

01

Create the first Latin Square

Let's create the first Latin square, \( L_1\), in standard form directly. It can be the identity matrix for simplicity. This means the diagonals are all filled with '1's and the remaining elements are cyclic permutations of the numbers 2 to 7.
02

Create the second Latin Square

Create the second Latin square, \( L_2\), orthogonal to \( L_1\). Start with an identity matrix as before, then permute the rows (except the first one) that every symbol pair is different.
03

Create the third Latin Square

Create the third Latin square, \( L_3\), orthogonal to both \( L_1\) and \( L_2\). Place the first row as 1,2,3,4,5,6,7 then place the subsequent numbers such that no pair is repeated in the orthogonal pairs of \(L_1\) and \(L_2\). The success of this step may depend on choices made in step 2, and sometimes a new \(L_2\) must be found.
04

Standardise the Latin Squares

At this point, \( L_1\) is already in standard form. Make \( L_2\) and \( L_3\) standard by permuting the rows and columns such that the first row and first column are in natural order.

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Most popular questions from this chapter

In a programming class Professor Madge has a total of \(n\) students, and she wants to assign teams of \(m\) students to each of \(p\) computer projects. If each student must be assigned to the same number of projects, (a) in how many projects will each individual student be involved? (b) in how many projects will each pair of students be involved?

Given a field \(F\), let \(f(x) \in F[x]\) where \(f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}\). a) Prove that \(x-1\) is a factor of \(f(x)\) if and only if \(a_{n}+a_{n-1}+\cdots+a_{2}+a_{1}+a_{0}=0\). b) If \(n\) is even, prove that \(x+1\) is a factor of \(f(x)\) if and only if \(a_{n}+a_{n-2}+\cdots+\) \(a_{2}+a_{0}=a_{n-1}+a_{n-3}+\cdots+a_{3}+a_{1}\).

Let \(s(x)=x^{4}+x^{3}+1 \in \mathbf{Z}_{2}[x]\). a) Prove that \(s(x)\) is irreducible. b) What is the order of the field \(\mathbf{Z}_{2}[x] /(s(x))\) ? c) Find \(\left[x^{2}+x+1\right]^{-1}\) in \(\mathbf{Z}_{2}[x] /(s(x))\). (Hint; Find \(a, b, c, d \in \mathbf{Z}_{2}\) so that \(\left[x^{2}+x+1\right]\) \(\left.\left[a x^{3}+b x^{2}+c x+d\right]=[1] .\right)\) d) Determine \(\left[x^{3}+x+1 \mid\left[x^{2}+1\right]\right.\) in \(\mathbf{Z}_{2}[x] /(s(x))\).

\text { Given a }(v, b, r, k, \lambda) \text {-design with } b=v \text {, prove that if } v \text { is even, then } \lambda \text { is even. }

a) Find all roots of \(f(x)=x^{2}+4 x\) if \(f(x) \in \mathbf{Z}_{12}[x]\). b) Find four distinct linear polynomials \(g(x), h(x), s(x), t(x) \in \mathbf{Z}_{12}[x]\) so that \(f(x)=\) \(g(x) h(x)=s(x) r(x)\) c) Do the results in part (b) contradict the statements made in the paragraph following Example 17.7?
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