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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

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 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 ω of 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:

i 0 1 2 3 4 5 6 7 8 9 ωi 1 2 4 8 5 10 9 7 3 6

The field table for F16 = F 42 :

i 0 1 2 3 4 5 6 7 8 ωi 0001 0010 0100 1000 0011 0110 1100 1011 0101

i 9 10 11 12 13 14 ωi 1010 0111 1110 1111 1101 1001

Paper 30

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2 Let C be 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 x and x′ of length n, w(x+x′) = w(x)+w(x′)−2w(x∧x′), where (x ∧ x′)j = xjx′j, 1 6 j 6 n.]

Prove that if the generating matrix G of C has no zero column then the total weight∑ x∈C

w(x) equals n2k−1.

[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 As and A⊥s denote the number of words of weight s in CH,` and C⊥H,`, respectively, with A0 = A⊥0 = 1 and A1 = A2 = 0. Check that

A3 = n(n− 1)

3! , A4 =

n(n− 1)(n− 3) 4!

, A5 = n(n− 1)(n− 3)(n− 7)

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 As in terms of Ks(2`−1), the value of the Kravchuk polynomial:

Ks(2`−1) = s∧2`−1∑

j=0∨s+2`−1−2`+1

( 2`−1

j

) ( 2` − 1− 2`−1

s− j

) (−1)j .

Here 0∨ s+2`−1 − 2` +1 = max [0, s+2`−1 − 2` +1] and s∧ 2`−1 = min [s, 2`−1]. Check that your formula gives the right answer for s = n = 2` − 1.

Paper 30 [TURN OVER

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3 Let ω be a root of m(X) = X5 + X2 + 1 in F32; given that m(X) is a primitive polynomial for F32, ω is a primitive (31, F32) root of unity. Use elements ω, ω2, ω3, ω4 to construct a binary narrow sense primitive BCH code X of length 31 and designed distance 5. Identify the cyclotomic coset {i, 2i, . . . , 2d−1i} for each of ω, ω2, ω3, ω4. Check that ω and ω3 suffice as defining zeros of X and that the actual minimum distance of X equals 5. Show that the generator polynomial g(X) for X is 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) = ω3 and 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 = F 52 and the list of irreducible polynomials of degree 5 over F2 are also provided to help with your calculations.]

The field table for F32 = F 52 :

i 0 1 2 3 4 5 6 7 8 ωi 00001 00010 00100 01000 10000 00101 01010 10100 01101

i 9 10 11 12 13 14 15 16 17 ωi 11010 10001 00111 01110 11100 11101 11111 11011 10011

i 18 19 20 21 22 23 24 25 26 ωi 00011 00110 01100 11000 10101 01111 11110 11001 10111

i 27 28 29 30 ωi 01011 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 = 2s − 1. If 2sl < ∑

06i6l+1

( n i

) then the binary narrow-sense

primitive BCH code of designed distance 2l + 1 has minimum distance 2l + 1.

Paper 30