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
3
2Let Cbe a binary linear [n, k] code and Cev the set of words x∈ C of even weight.
Prove that either (i) C=Cev or (ii) Cev is an [n, k 1] linear subcode of C.
[Hint: For binary words xand x0of length n,w(x+x0) = w(x)+w(x0)2w(xx0),
where (xx0)j=xjx0
j,16j6n.]
Prove that if the generating matrix Gof Chas no zero column then the total weight
P
x∈C
w(x) equals n2k1.
[Hint: Consider the contribution from each column of G.]
Denote by CH,` the binary Hamming code of length n= 2`1 and by C
H,` the dual
simplex code, `= 3,4, . . .. Is it always true that the n-vector (1, . . . , 1) (with all digits
one) is a codeword in CH,`? Let Asand A
sdenote the number of words of weight sin
CH,` and C
H,`, respectively, with A0=A
0= 1 and A1=A2= 0. Check that
A3=n(n1)
3! , A4=n(n1)(n3)
4! , A5=n(n1)(n3)(n7)
5! .
Prove that A
2`1= 2`1 (i.e., all non-zero words x∈ C
H,` have weight 2`1). By using
the last fact and the Mac Williams identity for binary codes, give a formula for Asin
terms of Ks(2`1), the value of the Kravchuk polynomial:
Ks(2`1) =
s2`1
X
j=0s+2`12`+1 2`1
j 2`12`1
sj(1)j.
Here 0 s+ 2`12`+ 1 = max [0, s + 2`12`+ 1] and s2`1= min [s, 2`1]. Check
that your formula gives the right answer for s=n= 2`1.
Paper 30 [TURN OVER
4
3Let ωbe a root of m(X) = X5+X2+ 1 in F32 ; given that m(X) is a primitive
polynomial for F32,ωis a primitive (31,F32) root of unity. Use elements ω,ω2,ω3,ω4to
construct a binary narrow sense primitive BCH code Xof length 31 and designed distance
5. Identify the cyclotomic coset {i, 2i, . . . , 2d1i}for each of ω,ω2,ω3,ω4. Check that ω
and ω3suffice as defining zeros of Xand that the actual minimum distance of Xequals
5. Show that the generator polynomial g(X) for Xis the product
(X5+X2+ 1)(X5+X4+X3+X2+ 1)
=X10 +X9+X8+X6+X5+X3+ 1.
Suppose you received a word u(X) = X12 +X11 +X9+X7+X6+X2+ 1 from a
sender who uses code X. Check that u(ω) = ω3and u(ω3) = ω9, argue that u(X) should
be decoded as
c(X) = X12 +X11 +X10 +X9+X7+X6+X2+ 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 F32 =F5
2and the list of irreducible polynomials of
degree 5over F2are also provided to help with your calculations.]
The field table for F32 =F5
2:
i012345678
ωi00001 00010 00100 01000 10000 00101 01010 10100 01101
i9 10 11 12 13 14 15 16 17
ωi11010 10001 00111 01110 11100 11101 11111 11011 10011
i18 19 20 21 22 23 24 25 26
ωi00011 00110 01100 11000 10101 01111 11110 11001 10111
i27 28 29 30
ωi01011 10110 01001 10010
The list of irreducible polynomials of degree 5 over F2:
X5+X2+ 1, X5+X3+ 1, X5+X3+X2+X+ 1,
X5+X4+X3+X+ 1, X5+X4+X3+X2+ 1;
they all have order 31. Polynomial X5+X2+ 1 is primitive.
Theorem. Let n= 2s1. If 2sl <P
06i6l+1 n
ithen the binary narrow-sense
primitive BCH code of designed distance 2l+ 1 has minimum distance 2l+ 1.
Paper 30
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