Virtual Labs

Reed-Muller Codes : Generator and Parity Check Matrices and the Minimum Distance

1. What is the

2 \times 2

binary kernel matrix

G_2

that is used to construct the generator matrix for all Reed-Muller codes?

\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

Explanation

\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}

Explanation

\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}

Explanation

\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

Explanation

2. Which matrix operation is repeatedly applied to the kernel

G_2

to create the full

2^m \times 2^m

matrix

G_2^{\otimes m}

a: Matrix Multiplication Explanation

Explanation

b: Kronecker Product (

\otimes

) Explanation

Explanation

c: Hadamard Product (Component-wise) Explanation

Explanation

d: Matrix Addition Explanation

Explanation

3. How is the generator matrix for the repetition code

RM(0, m)

formed from

G_2^{\otimes m}

a: By selecting all rows of

G_2^{\otimes m}

. Explanation

Explanation

b: By selecting all rows with Hamming weight equal to 1. Explanation

Explanation

c: By selecting only the first row (the all-ones vector). Explanation

Explanation

d: By selecting only the last row. Explanation

Explanation

4. How is the generator matrix for the full space

RM(m, m)

formed from

G_2^{\otimes m}

a: By selecting all rows, as

G_{RM(m,m)} = G_2^{\otimes m}

. Explanation

Explanation

b: By selecting only the first row. Explanation

Explanation

c: By selecting all rows with weight less than 2. Explanation

Explanation

d: By selecting all rows with even Hamming weight. Explanation

Explanation

5. What is the general rule for selecting rows from

G_2^{\otimes m}

to form the generator matrix for

RM(r, m)

a: Select all rows with Hamming weight

\le 2^{m-r}

. Explanation

Explanation

b: Select all rows with Hamming weight

\ge 2^{m-r}

. Explanation

Explanation

c: Select all rows with Hamming weight

\le r

. Explanation

Explanation

d: Select all rows with Hamming weight

\ge 2^r

. Explanation

Explanation

6. The rows of

G_2^{\otimes m}

are the evaluation vectors of a set of polynomials. What set of polynomials do they represent?

a: All multilinear monomials of degree exactly

r

. Explanation

Explanation

b: All multilinear monomials in

m

variables (e.g.,

1, X_1, X_2, X_1X_2, \dots

). Explanation

Explanation

c: All polynomials of total degree at most

r

. Explanation

Explanation

d: Only the single variables

X_m, \dots, X_1

. Explanation

Explanation

7. According to the duality theorem, what is the dual code of

RM(r, m)

, denoted

[RM(r, m)]^{\perp}

RM(m-r, m)

Explanation

RM(r, m)

(it is always self-dual) Explanation

Explanation

RM(m-r-1, m)

Explanation

RM(r-1, m)

Explanation

8. The inductive proof for minimum distance relies on the

(u|u+v)

construction. This implies

G_{RM(r,m)}

has a specific recursive block structure. What is it?

\begin{pmatrix} G_{RM(r-1, m-1)} & G_{RM(r-1, m-1)} \\ \mathbf{0} & G_{RM(r, m-1)} \end{pmatrix}

Explanation

\begin{pmatrix} G_{RM(r, m-1)} & G_{RM(r-1, m-1)} \\ G_{RM(r-1, m-1)} & G_{RM(r, m-1)} \end{pmatrix}

Explanation

\begin{pmatrix} G_{RM(r, m)} & \mathbf{0} \\ \mathbf{0} & G_{RM(r-1, m)} \end{pmatrix}

Explanation

\begin{pmatrix} G_{RM(r, m-1)} & G_{RM(r, m-1)} \\ \mathbf{0} & G_{RM(r-1, m-1)} \end{pmatrix}

Explanation