M. Phil. in Statistical Science Exam: Paper 36 - Algebraic Coding, Exams of Statistics

Download M. Phil. in Statistical Science Exam: Paper 36 - Algebraic Coding and more Exams Statistics in PDF only on Docsity! M. PHIL. IN STATISTICAL SCIENCE Monday 9 June 2003 1.30 to 3.30 PAPER 36 ALGEBRAIC CODING 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. You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 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−12 ] 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, n2 ] code X exists for each even n. Conversely, prove that if a binary linear [n, k] code X is self-dual then k = n2 . 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 = ps 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 Fsp with ps 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 F9, identifying the powers βi of a primitive element β ∈ F9 as vectors over F3. Indicate all vectors α in this table such that α4 = 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, F9) roots of unity as vectors over F3. Paper 36