# Algebraic Coding - Math Tripos - Past Exam, Exams for Mathematics. Agra University

## Mathematics

Description: This is the Past Exam of Math Tripos which includes Category Theory, Black Holes, Klein-Gordon Equation, Banach Algebras, Astrophysical Fluid Dynamics etc. Key important points are: Algebraic Coding, Reed–Solomon Codes, Generator Polynomial, Primitive Element, Field Table, Binary Linear Code, Generating Matrix, Binary Hamming Code of Length, Contribution From, Mac Williams Identity
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MATHEMATICAL TRIPOS Part III
Monday 3 June 2002 1.30 to 3.30
PAPER 30
ALGEBRAIC CODING
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
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
2
1Deﬁne 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 g1(X) over F16 for this code. Use the provided F16 ﬁeld table to
write g1(X) in the form ωi0+ωi1X+ωi2X2+. . ., identifying each coeﬃcient as a single
power of a primitive element ωof F16 .
Find the generator polynomial g2(X) and the minimum distance of a Reed–Solomon
code of length 10 and rank 6. Use the provided F11 ﬁeld table to write g2(X) in the form
a0+a1X+a2X2+. . ., where each coeﬃcient is a number from {0,1, . . . , 10}.
Determine a two-error correcting Reed–Solomon code over F16 and ﬁnd its length,
rank and generator polynomial.
The ﬁeld table for F11 ={0,1,2,3,4,5,6,7,8,9,10}, with addition and multiplica-
tion mod 11: i012345 6789
ωi12485109736
The ﬁeld table for F16 =F4
2:
i012345678
ωi0001 0010 0100 1000 0011 0110 1100 1011 0101
i9 10 11 12 13 14
ωi1010 0111 1110 1111 1101 1001
Paper 30
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