Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
The third paper of the m. Phil. In statistical science exam focused on algebraic coding. The paper includes definitions, proofs, and exercises on topics such as dual codes, self-dual codes, finite fields, roots of unity, and cyclic codes. Candidates are required to attempt three questions out of four, bringing allowed materials for assistance. The paper covers important concepts in error correction and coding theory.
Typology: Exams
1 / 3
This page cannot be seen from the preview
Don't miss anything!
Monday 9 June 2003 1.30 to 3.
Attempt THREE questions.
There are four questions in total.
The questions carry equal weight.
Candidates may bring into the examination any lecture notes made during the course,
printed lecture notes, example sheets and model solutions,
and books or their photocopies.
1 Define the dual X
⊥ of a linear [n, k] code of length n and dimension k with alphabet
F. Prove or disprove that if X is a binary [n,
n− 1 2
] code with n odd then X
⊥ is generated
by a basis of X plus the word 1... 1. Prove or disprove that if a binary code X is self-dual:
X = X
⊥ then n is even and the word 1... 1 belongs to X.
Prove that a binary self-dual linear [n,
n 2
] code X exists for each even n. Conversely,
prove that if a binary linear [n, k] code X is self-dual then k =
n 2
Give an example of a non-binary linear self-dual code. Justify your answer.
2 Define a finite field Fq with q elements and prove that q must have the form q = p
s
where p is prime integer and s > 1 positive integer. Check that p is the characteristic of
Fq.
Prove that for any p and s as above there exists a finite field F
s p with^ p
s elements,
and this field is unique up to isomorphism.
Prove that the set F∗ ps of the non-0 elements of Fps is a cyclic group Zps− 1.
Write the field table for F 9 , identifying the powers β
i of a primitive element β ∈ F 9
as vectors over F 3. Indicate all vectors α in this table such that α
4 = e.
3 What is an (n, Fq )-root of unity? Show that the set E
(n,q) of the (n, Fq )-roots of
unity form a cyclic group. Check that the order of E(n,q)^ equals n if n and q are co-prime.
Find the minimal s such that E
(n,q) ⊂ Fqs^.
Define a primitive (n, Fq )-root of unity. Determine the number of primitive (n, Fq )-
roots of unity when n and q are co-prime. If ω is a primitive (n, Fq )-root of unity, find the
minimal such that ω ∈ Fq.
Find all (4, F 9 ) roots of unity as vectors over F 3.
Paper 36
