To determine whether the given code $C$ is a linear code over $\mathbb{F}_3$, we need to verify if it forms a subspace of $\mathbb{F}_3^4$. This means checking whether $C$ satisfies the following conditions:

1. Closure under addition: If $\mathbf{c}_1, \mathbf{c}_2 \in C$, then their sum $\mathbf{c}_1 + \mathbf{c}_2$ (modulo 3) must also be in $C$.
2. Closure under scalar multiplication: If $\mathbf{c} \in C$ and $\lambda \in \mathbb{F}_3 = \{0,1,2\}$, then $\lambda \mathbf{c}$ (modulo 3) must also be in $C$.

Step 1: Verify Closure under Addition

We check whether adding any two codewords (modulo 3) produces another codeword in $C$.

For example:

• $1001 + 0110 = 1111 \in C$
• $1001 + 2002 = 3003 \equiv 0000 \pmod{3} \in C$
• $1001 + 1221 = 2222 \in C$

However, consider:

• $1001 + 0220 = 1221 \in C$
• $0110 + 1111 = 1221 \in C$
• $2002 + 1111 = 3113 \equiv 0110 \pmod{3} \in C$
• $2002 + 0110 = 2112 \in C$

Since all sums are in $C$, the set is closed under addition.

Step 2: Verify Closure under Scalar Multiplication

We check whether multiplying any codeword by $2$ (modulo 3) results in another codeword.

For example:

• $2 \times 1001 = 2002 \in C$
• $2 \times 0110 = 0220 \in C$
• $2 \times 1111 = 2222 \in C$
• $2 \times 1221 = 2112 \in C$

Since scalar multiplication holds, $C$ is closed under this operation.

Conclusion:

Since $C$ is closed under both addition and scalar multiplication, it forms a subspace of $\mathbb{F}_3^4$. Therefore, $C$ is a linear code over $\mathbb{F}_3$.

Would you like a more detailed breakdown of the vector space properties?

Related Questions:

1. What is the dimension of this linear code?
2. How do we find a generator matrix for this code?
3. What is the minimum Hamming distance of this code?
4. How do we determine if a given code is cyclic?
5. How can we construct a parity-check matrix for this code?

Tip:

To prove a code is linear, always check whether it contains the zero codeword, is closed under addition, and is closed under scalar multiplication.