First written: March 17, 2024
This version: April 11, 2024
The class should be partitioned into teams with optimal size of 3, but groups of size 1,2, and 4 are also OK. Each team will have a leader. You are welcome to volunteer to be a leader, first-come first-serve. Otherwise I will pick a leader at random.
Students should pick a project no later then 8:00 pm, March 31, 2024 (it should be included in the homework due that day). But the membeship is first-come-first-serve, so you are welcome to email me as soon as you decide, in order to guarantee a place.
The team leader will be in charge of coordinating the various contributions, and writing up (with the team's help) the first draft of a paper that will be definitely posted here. In some cases, after more work, some of the projects may lead to a "real" paper.
The projects should be ready by: Wed., May 1, 2024, 11:00pm emailed to ShaloshBEkhad@gmail.com Note earlier deadline.
The student presentation will take place, via WebEx, on May 2, 2024, 12:00-2:30pm (WebEx invitation will be sent by May 1)
In additon to the text file with Maple source code called projX.txt the team leader should also submit a file projX.pdf with a write-up of the approach, with summarizing the data (if applicable) and/or the puzzle book (if applicable)
For as large N as possible, and as large A as possible, and as large Q as possible, find all cyclic [n,n-a] codes over GF(q) for n ≤ N, a ≤ A, and prime q ≤Q. Since a is much smaller (usually) then n-a, you should use the MacWilliams identity (see Chapter 13) (rediscovered by Joseph Koutsoutis).
For each such code, find the average, and standard-deviation of the weight.
(You may use procedures AveAndMoms(f,x,N) and Alpha(f,x,N) in the little file Stat.txt)
Also compare it to what you would get from random [n,n-a] codes over GF(q) using Dr. Z.'s paper and Maple package available from here
Project Leader: Lucy Martinez
Other team member: Himanshu Chandra, Nkhalo Malawo
For as large Q as possible, find the weight enumerators of BCH codes (defined on p. 131 of the book) with
d=2t+1 ≤ n ≤ q-1
for all primes ≤ Q.
For each such code, find the average, and standard-deviation of the weight.
(you may use (procedures AveAndMoms(f,x,N) and Alpha(f,x,N) in the little file Stat.txt)
Also compare it to what you would get from random [n,n-(d-1)] codes over GF(q) using Dr. Z.'s paper and Maple package available from here
Project Leader and only member: Joseph Koutsoutis
Given a word w=[w[1],w[2],...,w[n]] and v=[v[1],v[2],...,v[m]], we say that the word v follows the word w, if a tail of size at least 2 of w is the head of of v. i.e. there exists a k &ge 2 such that
[v1,..., vk]=[w[n-k+1], ..., w[n]]
For example vermont follows love, and ontario follows vermont.
A word-chain of length r is a list of words
[w[1],w[2], ..., w[r]] such that w[i+1] follows w[i] (but w[i+2] does not follow w[i], i.e. the overlaps are disjoint)
It is a word-cycle if, in addition w[1] follows w[r]
Using Dr. Z.'s Maple package CRYPTOGRAM.txt that uses the data-base of words (in Maple format) ENGLISH.txt construct a data base of all word-chains and word-cycles of as large a length as possible.
Using Maple's graph, create a "snake charmer" puzzle-book, inspired by Patrick Berry's ingenious puzzles, published in the New York Times magazine. Here is an example from March 17, 2024.
Project Leader: Ramesh Balaji
Other team member: Daniel Elwell, Nuray Kutlu
A word v follows a word w if it is obtained by inserting one letter somewhere (including at the beginning or end)
Using Dr. Z.'s Maple package CRYPTOGRAM.txt that uses the data-base of words (in Maple format) ENGLISH.txt construct a data base of all word pyramids possible.
By adding definitions, use the best of them to create a puzzle book. Where you are given the first and last words of the pyramid and the solver has to figure out the intermedia words. This "follower" relation is given by procedure FOllowers(W).
Another type of of chain (not a pyramid) where one word follows another by changing one letter (e.g. [b,a,t] -> [b,e,t]). This is given by procedure Followers1(W). Make sure that there are no repeats.
Project Leader: Isaac Lam
Other team members: Dayoon Kim
Project Leader: Pablo Blanco Hinojosa
Other team members: Aurora Hiveley and Kaylee Weatherspoon
First use wikipedia to get the lists of
The following words are anagrams of US state capitals, can you find them?
Project Leader: Alex Varjabedian
Other team members: Ryan Badi and Shaurya Baranwal
Given a large positive integer A Consider the sequences of the form
Coefficient of x^0 in (a+b/x+c*x)^n for -A ≤ a,b,c ≤ A, and igcd(a,b,c)=1
Construct a data base, in the style of the OEIS, that lists the first 30 terms of each, and the linear recurrences obtained via the famous Almkvist-Zeilberger algorithm, with the command
AZd( (a+b/x+c*x)^n/x,n,N)[1]
in the Maple package EKHAD.txt
Find out which ones are already in the OEIS.
Project Leader (and only member): Gloria Liu
Project Leader (and only member) Alex Valentino
[Project suggested by Alex Valentino]
Project Leader: Robert Dougherty-Bliss
Other team member: George Spahn
[Project suggested by RBD and GS]