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Highlighted topic is coming up soon. |
Date Planned |
Date Actual | Topic |
| We will review groups as it becomes necessary, but you may want to study Chapter 3 of Applied Abstract Algebra on your own. | ||
| Week 1 | Weeks 1-2 | Syllabus. Ch. 4: Coding Theory, § 16 (Introduction to Coding Theory). The idea of Coding Theory. A discussion of Finite Fields. The Hamming distance and various bounds. |
| Weeks 2-3 | Weeks 2-5 | Ch. 4: Coding Theory, § 17 (Linear Codes). The parity check and generator matrices. A refresher on Cosets of Subgroups. Decoding, correcting errors. Bounds on the size of a linear code. |
| Weeks 3-4 | Alas. | Ch. 4: Coding Theory, § 18 (Cyclic Codes). A discussion of polynomial rings, especially Ideals and Quotient rings. Criterion for a code to be cyclic. Generator and check polynomials. A discussion of primitive roots of unity. Decoding cyclic codes. |
| Week 5 | Week 6 | Ch. 5: Cryptology, § 21 (Classical Cryptosystems). The idea of encryption. The cipher that cost Cæsar a battle, with a generalization or ten (affine modular, Vigenère, one-time pads, matrix methods such as the Hill cipher, and permutation polynomials). Dangers and disadvantages of classical encryption. |
| Week 6 | Ch. 5: Cryptology, § 22 (Public Key Cryptosystems). | |
| Week 7 | Ch. 5: Cryptology, § 23 (Discrete Logarithms and Other Ciphers). | |
| Test 1 | ||
| Week 8 | Hackenbush The basic relationship between finite games and finite numbers. Positive, negative, and rational numbers. | |
| Week 9 | Nim Impartial games, *-games, subtraction games. | |
| Week 10 | Ideal Nim Infinite games, and infinite numbers. | |
| Test 2 | ||
| Week 11 | Computational algebra, pt. 1 Fraction-free determinant computation. Gaussian elimination, Bareiss' method, Dodgson's Method, Leggett's Method. | |
| Week 12 | Computational algebra, pt. 2 Division-free determinant computation. Black box methods. | |
| Week 13 | Computational algebra, pt. 3 Gröbner bases. The Macaulay matrix. Traditional methods to compute Gröbner bases. | |
| Week 14 | Computational algebra, pt. 4 Gröbner bases. Signature-based methods. Dynamic algorithms. | |
| Test 3 | ||
| Week 15 - until | Student presentations | |