Math 3205辅导、data讲解、Java/c++，Python程序语言讲解 讲解SPSS|解析Haskell程序

日期：2020-05-12 08:37

Math 3205 Assignment 5 December 2019
Clearly write your answers to the questions showing all reasons, working and checks and indicate what
each mathematical calculation is doing. Do not erase anything. Include all rough work and do not commit
plagiarism. Feel free to write explanations of what you are thinking at each stage, nothing you can write
can lose you marks!
1. Let the addition table in Z2[β] where β
3 = β + 1 be as follows:
(a) Count how many latin squares there are within L of each cardinality (and explain why none exist
for some cardinalities) from addition of subsets of the elements {0, 1, β, β + 1, β2
, β2 + 1, β2 +
β, β2 + β + 1}, keeping them in this order. [no need to count all permutations of them] [4]
Francis Illya Xucheng Mingxin Aidan
α β β + 1 β2 β2 + 1 β2 + β
(b) Using your randomly chosen element α find the 8×8 latin square M which is formed using sα+t
for the (s, t) entry using the elements. Do not just fill out the matrix, explain your process as you
go along. [4]
(c) Verify that M is orthogonal to L by highlighting all elements in L which are in the same place
as each of the elements of M in a different colour/shape/pattern and verify that the expected
behaviour happens which proves orthogonality. [2]
2. (a) Choose a prime number q congruent to 3(mod 4) (different from anyone else in the class you have
talked to) and find the squares mod q, use them to form the Paley design Pq for q, and verify that
any number appears in the same number of blocks with all other numbers. [2]
(b) Find the parameters v, b and r for Pq, give the incidence matrix and verify that k and λ satisfy
the equations relating the parameters. [3]
(c) Create the Hadamard Matrix H corresponding to Pq and verify that HT H is a multiple of the
identity matrix, and list the 3-design based on it, verifying how many times a triple occurs. [3]
(d) Choose a block c of Pq and form either the derived or residual design with respect to c and
calculate the parameters. [2]