Given that K = 3 and the transmitted message is 011001100100001, what is the original code?

To find the original code, we need to understand the context of \( K = 3 \). In coding theory, \( K \) could represent the length of the message before encoding, the number of bits in a block for error correction, or a similar parameter. Since the problem doesn't specify, I'll assume that \( K = 3 \) refers to a simple repetition code, where each bit of the original message is repeated 3 times to form the transmitted message.

Given the transmitted message is `011001100100001`, we'll decode it by taking every 3 bits and identifying the majority bit (which is the bit that appears most frequently within those 3 bits). This process is based on the assumption that the repetition code is used to correct single-bit errors.

Let's break down the transmitted message:

1. **011** → Majority bit: **1**
2. **001** → Majority bit: **0**
3. **100** → Majority bit: **0**
4. **100** → Majority bit: **0**
5. **001** → Majority bit: **0**
6. **001** → Majority bit: **0**

So, the original code (before encoding) is `10000`.

### Do you want details or have any questions?

Here are five questions to explore further:

1. What if the transmitted message had more errors? How would it affect decoding?
2. Can you explain what other values of \( K \) could represent in different coding schemes?
3. What are some common types of error-detecting and error-correcting codes?
4. How does the repetition code compare to more sophisticated error-correcting codes like Hamming codes?
5. How can the concept of majority voting be applied in other areas of computing or mathematics?

**Tip:** When decoding error-correcting codes, it's essential to understand the specific code structure and error tolerance to apply the correct decoding method.

Understand how to decode a transmitted message with K = 3 using coding theory principles. Explore the application of repetition codes and majority voting in error correction. Detailed solution provided with step-by-step decoding process.

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