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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