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(x∧x0),

where (x∧x0)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 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 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(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 Asin

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

Ks(2`−1) =

s∧2`−1

X

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

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, . . . , 2d−1i}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= 2s−1. 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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