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Theory

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Procedure

Simulation

Posttest

References

Contributors

Feedback

Majority Logic Decoding of Reed-Muller Codes

Choose difficulty:

Beginner

Intermediate

Advanced

1. Let

f(X_1, X_2) = X_1X_2 + X_2

be a Boolean polynomial over

\mathbb{F}_2

. What is

f(1, 0)

a: 0 Explanation

Explanation

b: 1 Explanation

Explanation

c: 2 Explanation

Explanation

X_1

Explanation

2. Which of the following best explains why the generator matrix of

RM(1, m)

has rank

m + 1

a: Because there are

m

linearly independent monomials of degree 1 and one constant term. Explanation

Explanation

b: Because there are

2^m

possible evaluation points. Explanation

Explanation

c: Because each codeword is orthogonal to its complement. Explanation

Explanation

d: Because

RM(1,m)

is self-dual. Explanation

Explanation

3. Let

RM(2,3)

denote the second-order Reed–Muller code of length 8. Which of the following is *not* a basis monomial of this code?

X_1X_2

Explanation

X_2X_3

Explanation

X_1X_2X_3

Explanation

X_1

Explanation

4. If a codeword of

RM(2,4)

is obtained by evaluating

f(X_1,X_2,X_3,X_4) = X_1X_2 + X_3X_4 + 1

, how many coordinates equal 1 in this codeword?

a: 4 Explanation

Explanation

b: 6 Explanation

Explanation

c: 8 Explanation

Explanation

d: 10 Explanation

Explanation

5. A specific Reed-Muller code

C = RM(r, 5)

is known to have a minimum distance

d_{min} = 8

. What is the order

r

of this code?

r = 1

Explanation

r = 2

Explanation

r = 3

Explanation

r = 4

Explanation

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