MTH351 Tutor-Marked Assignment 01

This assignment is worth 20% of the final mark for MTH351 – Coding Theory.

Learning Outcome:

● Show the existence/non-existence of certain codes with certain parameters.
● Demonstrate the decoding of BCH codes, Reed-Solomon codes and Quadratic-residue codes.
● Calculate generator matrix and parity-check matrix of a given linear code.
● Construct certain linear codes from other given linear code.
● Determine the generator polynomial of a given cyclic code.
● Compute the capacity of a discrete memoryless communications channel.

What the MTH351 Tutor-Marked Assignment 01 asks you to do

MTH351 Coding Theory studies how information can be encoded so that errors introduced by a noisy channel can be detected and corrected. The course works over finite fields and covers linear codes, cyclic codes, bounds on code parameters and important families such as BCH, Reed-Solomon and quadratic-residue codes.

This TMA01 is worth 20% of the final mark for MTH351 – Coding Theory. It tests whether you can prove facts about codes, carry out decoding and matrix calculations accurately, construct new codes from existing ones and compute channel capacity. Clear working and correct reasoning matter as much as the final answer.

MTH351 learning outcomes explained

Each outcome below tells you what kind of working the marker expects.

How to structure your MTH351 TMA01

1. Read each question carefully and note the field, code length, dimension and minimum distance.
2. State the definitions and theorems you will use before applying them.
3. Show full working: finite field arithmetic, matrix row operations and polynomial factorisation.
4. Check your results, for example by verifying codewords satisfy the parity checks.
5. Explain proofs in words as well as symbols so the logic is easy to follow.
6. Present answers neatly with clear labels for each part and final answers highlighted.

Key concepts to revise for MTH351

• Finite fields GF(q) – Arithmetic in prime and extension fields, primitive elements and minimal polynomials.
• Linear codes [n, k, d] – Generator and parity-check matrices, standard form and syndrome decoding.
• Bounds on codes – Hamming, Singleton and Gilbert-Varshamov bounds for existence questions.
• Cyclic codes – Generator and check polynomials, and factorising x^n − 1.
• BCH and Reed-Solomon codes – Designed distance, syndromes and error-locator polynomials.
• Quadratic-residue codes – Construction from quadratic residues modulo a prime and their decoding.
• Code constructions – Extending, puncturing, shortening and dual codes, and their effect on parameters.
• Channel capacity – Entropy, mutual information and capacity of channels such as the binary symmetric channel.

Common mistakes to avoid

• Making arithmetic slips in finite fields, especially in extension fields.
• Giving final answers with no working, which loses method marks.
• Forgetting to verify that a generator and parity-check matrix are consistent.
• Mixing up the parameters of a code after puncturing or shortening.
• Using natural logarithms instead of base 2 when computing capacity in bits.

MTH351 TMA01 FAQs

What is the MTH351 TMA01 about?

It covers existence of codes, decoding of BCH, Reed-Solomon and quadratic-residue codes, generator and parity-check matrices, code constructions, cyclic codes and channel capacity.

How should I present my MTH351 TMA01?

Answer each part in order, state the results you use, show full working and check your answers before submitting.

Which topics should I revise for this course?

Finite fields, linear and cyclic codes, bounds on codes, BCH and Reed-Solomon decoding, and channel capacity are the core areas.

Can I get help with my MTH351 TMA01?

Yes. Our Singapore-based subject tutors can guide you through the brief, working and presentation so you can submit with confidence.