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The questions and instructions for paper 30 of the mathematical tripos part iii algebraic coding exam held on 3 june 2002. The exam covers topics such as reed-solomon codes, binary linear codes, and bch codes. Candidates are allowed to bring lecture notes, printed materials, and books to the exam. Questions on defining and proving properties of reed-solomon codes, finding generator polynomials and minimum distances, and proving properties of binary hamming codes.
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Monday 3 June 2002 1.30 to 3.
Attempt THREE questions
There are three 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 Reed–Solomon codes and prove that they are maximum distance separable. Prove that the dual of a Reed–Solomon code is a Reed–Solomon code.
Find the minimum distance of a Reed–Solomon code of length 15 and rank 11 and the generator polynomial g 1 (X) over F 16 for this code. Use the provided F 16 field table to write g 1 (X) in the form ωi^0 + ωi^1 X + ωi^2 X^2 +.. ., identifying each coefficient as a single power of a primitive element ω of F 16.
Find the generator polynomial g 2 (X) and the minimum distance of a Reed–Solomon code of length 10 and rank 6. Use the provided F 11 field table to write g 2 (X) in the form a 0 + a 1 X + a 2 X^2 +.. ., where each coefficient is a number from { 0 , 1 ,... , 10 }.
Determine a two-error correcting Reed–Solomon code over F 16 and find its length, rank and generator polynomial.
The field table for F 11 = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 }, with addition and multiplica- tion mod 11: i 0 1 2 3 4 5 6 7 8 9 ωi^1 2 4 8 5 10 9 7 3
The field table for F 16 = F 24 :
i 0 1 2 3 4 5 6 7 8 ωi^0001 0010 0100 1000 0011 0110 1100 1011
i 9 10 11 12 13 14 ωi^1010 0111 1110 1111 1101
Paper 30
3 Let ω be a root of m(X) = X^5 + X^2 + 1 in F 32 ; given that m(X) is a primitive polynomial for F 32 , ω is a primitive (31, F 32 ) root of unity. Use elements ω, ω^2 , ω^3 , ω^4 to construct a binary narrow sense primitive BCH code X of length 31 and designed distance
(X^5 + X^2 + 1)(X^5 + X^4 + X^3 + X^2 + 1) = X^10 + X^9 + X^8 + X^6 + X^5 + X^3 + 1.
Suppose you received a word u(X) = X^12 + X^11 + X^9 + X^7 + X^6 + X^2 + 1 from a sender who uses code X. Check that u(ω) = ω^3 and u(ω^3 ) = ω^9 , argue that u(X) should be decoded as
c(X) = X^12 + X^11 + X^10 + X^9 + X^7 + X^6 + X^2 + 1
and verify that c(X) is indeed a codeword in X.
[You may quote, without proof, a theorem from the course (see below) but should check its conditions. The field table for F 32 = F 25 and the list of irreducible polynomials of degree 5 over F 2 are also provided to help with your calculations.]
The field table for F 32 = F 25 :
i 0 1 2 3 4 5 6 7 8 ωi^00001 00010 00100 01000 10000 00101 01010 10100
i 9 10 11 12 13 14 15 16 17 ωi^11010 10001 00111 01110 11100 11101 11111 11011
i 18 19 20 21 22 23 24 25 26 ωi^00011 00110 01100 11000 10101 01111 11110 11001 i 27 28 29 30 ωi^01011 10110 01001
The list of irreducible polynomials of degree 5 over F 2 : X^5 + X^2 + 1, X^5 + X^3 + 1, X^5 + X^3 + X^2 + X + 1,
X^5 + X^4 + X^3 + X + 1, X^5 + X^4 + X^3 + X^2 + 1;
they all have order 31. Polynomial X^5 + X^2 + 1 is primitive.
Theorem. Let n = 2s^ − 1. If 2 sl^ <
06 i 6 l+
n i
then the binary narrow-sense
primitive BCH code of designed distance 2 l + 1 has minimum distance 2 l + 1.
Paper 30
