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Reed-Muller Codes - Polynomial View of Encoding

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1. The polynomial

f(X_1, X_2) = 1 + X_1

over

\mathbb{F}_2

is used to generate a codeword. What is the codeword if the evaluation points are ordered as

(0,0), (0,1), (1,0), (1,1)

a: (1, 1, 0, 0) Explanation

Explanation

b: (1, 0, 1, 0) Explanation

Explanation

c: (0, 0, 1, 1) Explanation

Explanation

d: (1, 1, 1, 1) Explanation

Explanation

2. The polynomial

f(X_1, X_2, X_3) = X_1 + X_2X_3

is used to generate a codeword. To which Reed-Muller code does this codeword belong?

RM(1, 3)

Explanation

RM(2, 3)

Explanation

RM(3, 2)

Explanation

RM(2, 2)

Explanation

3. The codeword

(0, 1, 0, 1)

of length 4 is from a Reed-Muller code in variables

X_1, X_2

. Which polynomial generates this codeword? (Assume evaluation order is

(0,0), (0,1), (1,0), (1,1)

f(X_1, X_2) = X_1

Explanation

f(X_1, X_2) = X_2

Explanation

f(X_1, X_2) = X_1 + 1

Explanation

f(X_1, X_2) = X_1 + X_2

Explanation

4. What is the length of any codeword in the Reed-Muller code

RM(r, m)

2^r

Explanation

2^m

Explanation

\binom{m}{r}

Explanation

m+r

Explanation

5. Let

c_f = (0,0,1,1)

be the codeword for

f(X_1, X_2) = X_1

and

c_g = (0,1,0,1)

be the codeword for

g(X_1, X_2) = X_2

. What is the codeword corresponding to the polynomial

h(X_1, X_2) = f+g = X_1+X_2

a: (0,1,1,1) Explanation

Explanation

b: (0,0,0,0) Explanation

Explanation

c: (0,1,1,0) Explanation

Explanation

d: (1,1,1,1) Explanation

Explanation

6. What is the lowest order Reed-Muller code that contains the codeword

c = (0, 1, 0, 1, 0, 1, 1, 0)

RM(1, 3)

Explanation

RM(2, 3)

Explanation

RM(3, 3)

Explanation

RM(0, 3)

Explanation

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