Information and Resources for the Final Project
Read these articles on "how to give a good math talk" Article1, Article 2, Article 3, A Summary of Important Points. This web site is also very useful. It is specifically designed to help people give good math talks.
The first two articles are by the same authors, and the second one is an update on the first one (the first one was published in 1998 when most talks were by overhead transperencies). You may want to start with the second article.
Important Note: Please do not choose a topic without looking at resources first. Therefore, get the books (order them from Ohio link if necessary) well in advance of the first deadline.
Time Line
1. Project Description: Due date: Fri Apr 4 (W 10). First choose a topic in Coding Theory or Cryptography. Below are some suggestions (and resources) about possible topics, but you may choose a different topic. This one-page outline should include what you want to cover, and what sources you will be using.
2. Detailed Outline: Due date: Fri Apr 18 (W 12). This should probably be 3 pages or so (the more, the better), and should include precise statements of theorems, sketches of proofs, and summaries of contextual or historical material. Include references.
3. In-class presentations. During the last week of class
4. The final paper: Due Friday, May 9, 1:30 pm (can turn it in earlier)
Ideas About Possible Topics
Here is a list of some possible topics, with references as a starting point. You can also find a lot of resources on the internet.Your topic does not have to be one of these, you can choose a different topic. Some of these topics are covered in our textbook. If you choose a topic that is in the textbook, you must use other resources as well. In general, you should use several different sources. Accuracy and correctness are crucial. Give complete and careful references, do not plagiarize. The numbers for references refer to the list of resources at the end of the course syllabus. The material we are planning to cover from our textbook for the rest of the semester is as follows: Finish Chp 4, 10.1, 10.2, 11.1, Most of Chp 12, and if time permits parts of Chp 5 (first 2-3 sections). (This list and references may change/grow in the next couple of weeks)
Although not strictly required, you are encouraged to include computations or computer searches with Magma in your projects.
The list below is in no particular order.
Coding Theory
0. A computational project in Coding Theory using Magma. Contact the professor if you are interested in this.
1. Weight Enumerators and MacWilliam's identity. This is a well-known and fundamental theorem in coding theory relating weight enumerator of a code to its dual. Can be found in most coding theory books but there is a more accessible proof in [14]. You may include material on self-dual codes.
2. BCH Codes and their decodings. Can be found in most coding theory books.
3. Reed Solomon Codes and their decodings. Can be found in most coding theory books.
4. Codes and Designs (combinatorial designs). Steiner triple system from the Golay code (one of the many wonderful things about the Golay code) Start with [3] and [6].
5. Codes over Z_4. [3], [6], [9].
6. Convolutional Codes. [9], [6],[13],[15],[21],[27]
7. Low Density Parity Check (LDPC) Codes. [13]
8. Quadratic Residue Codes: [3],[6], [9],[10],[14].
9. Reed-Muller Codes. Can be found in most coding theory books.
10. Games and Codes, Lexicodes: [3],[22], and this paper. This project may involve a programming part to generate lexicodes.
11. You can find ideas about computational/programming projects in [24] (for both coding theory and cryptography).
12. Also see these web pages [22] for more ideas and tips for Mapla/Magma programming: 1, 2, 3
13. Exhaustive searches for cyclic or constacyclic codes over various fields. This paper, which is the outcome of the summer science research of Kenyon student in summer 2013, is a good one to start with. There is a possibility that you may discover record breaking codes (in that case you will be mentioned on the table of best known codes and became famous, at least in coding theory community!).
14. Quasi-cyclic and quasi-twisted codes. A generalization of cyclic codes that contain many good codes. Many of the best known codes are from this family. After describing their structure, it may be possible to search for new codes from this class and possibly find some new, record breakingcodes! (in that case you will be mentioned on the table of best known codes). Start with these papers 1, 2, (parts of the chapter) 3
15. Skew-cyclic codes. These are also a generalization of cyclic codes but they live in a non-commutative algebraic structure known as skew polynomial ring. These codes are recenty introduced in 1 and they are known to contain good codes 2. They are also related to quasi-cyclic codes 3.
16. There are other computational projects in coding theory (involving Magma). Talk to the professor if you are interested in doing a computational project.
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Cryptography
1. Secret Sharing Schemes. [1] and first edition of (1995) [11].
2. Cryptographic Hash Functions. [5],[11], [16],[17] http://csrc.nist.gov/CryptoToolkit/tkhash.html
3. Integer Factoring Algorithms: [1],[5],[11], [16].
4. Primality Tests and Prime Number Generation: [5], [11],[16],[17].
5. Elliptic Curve Cryptography: [1],[5]
6. Pseudo-Random Number Generation: [11], [16]. [17],[26].
7. Computational Complexity: [16],[18]
8. Quantum Computation and (Quantum Error-Correcting Codes or Quantum Cryptography): [1], [19],[23], [25].
9. NTRU: http://www.ntru.com
10. Knapsack Cryptosystems
11. History of Cryptography (be sure to include sufficient mathematical content). See the relevant articles on this page. There is also more there.
12. There are many possible topics and links on this page.
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Database
Search the Ohio Link for additional resources (or locating the resources listed above).
Use the search engine of MathSciNet to find articles on particular topics, or particular articles referenced in other articles.
Journals
Designs, Codes and Cryptography
Advances in Mathematics of Communication
One of the most important journals for information theory is IEEE Transactions on Information Theory. You don't have access to it. If you need any articles from that journal, let me know.