Module manager: TBC
Email: TBC
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2026/27
None
MATH3044 Number Theory MATH3153 Coding Theory
This module is not approved as a discovery module
Number theory explores the natural numbers. Central themes include primes and modular arithmetic, dating back at least to Euclid, and Diophantine equations as in Fermat's Last Theorem, finally proved only in the 1990s. It is a constantly developing field with many applications, such as in cryptography for the secrecy of digital data. The subject of error correcting codes is modern, starting with an article by Shannon in 1948. It concerns the practical problem of ensuring reliable transmission of digital data through a “noisy” channel. Applications include wireless communication, transmitting satellite pictures, designing registration numbers and storing data. The theory uses methods from algebra and combinatorics. The module introduces both subjects, emphasising the common features, such as modular arithmetic and finite fields, and applications to cryptography and error-correcting codes.
The module will introduce students to the foundations of number theory, focussing on primes, modular arithmetic, and finite fields, reaching into applications to cryptography. The module will also discuss error correcting codes. It will study error detection and correction, also from the point of view of probability. The focus will be on linear codes and syndrome decoding.
On successful completion of the module students will be able to: 1. State basic results in number theory and coding theory, reproduce short proofs, or find similar proofs. 2. Work with divisors, primes and prime factorisations, and use the Euclidean algorithm. 3. Compute in modular arithmetic and finite fields, including applying Fermat’s Little Theorem and Euler’s theorem. 4. Use primitive roots and other methods to test numbers for primality. 5. Explain encryption and decryption in the RSA cryptosystem. 6. Give examples of codes over prime fields. 7. Determine the minimum distance of a code and explain what this means for error correction and detection. 8. Determine the size of various sets associated to codes. 9. Compute the word error probability of a linear code and the probability of undetected errors. 10. Use the parity check matrix to determine properties of codes and to perform syndrome decoding.
Prime factorization and applications, Congruences and finite fields. · Fermat’s Little Theorem, Wilson’s Theorem, Chinese Reminder Theorem. · Euler’s Totient function, Primitive roots. · Public key cryptography (RSA), primality testing. · What is Coding Theory? (Data, Block encoding map, Block code, Error detection and correction, Probabilities) · Hamming distance (Minimum distance of a code, Equivalent codes) · Linear codes (Encoding, Standard array, Coset decoding, Probabilities of error correction and detection) · Dual Codes (Parity check matrix, syndrome decoding) · Hamming Codes Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar: · (Number theory) Pythagorean triples, sums of squares, Gaussian integers and generalisations and finite field extensions, continued fractions and Pell’s equation, quadratic reciprocity. · (Coding Theory) Bounds for the size of codes (Singleton and Ball Packing Bound), Golay codes.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 44 | 1 | 44 |
| Private study hours | 156 | ||
| Total Contact hours | 44 | ||
| Total hours (100hr per 10 credits) | 200 | ||
Formative feedback will be provided on regular example sets or other similar learning activity
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Last updated: 30/04/2026
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