| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To calculate the Hamming pairwise distances between the codewords, we need to compare each pair of codewords and count the number of positions at which the corresponding bits are different. The codewords are: 000000, 010101, 101010, 110110. Let's calculate the Hamming distances: 1. Between 000000 and 010101: Positions at which bits differ: 1, 2, 4, 5, 6 Hamming distance: 5 2. Between 000000 and 101010: Positions at which bits differ: 1, 3, 4, 6 Hamming distance: 4 3. Between 000000 and 110110: Positions at which bits differ: 1, 2, 3, 5, 6 Hamming distance: 5 4. Between 010101 and 101010: Positions at which bits differ: 1, 2, 3, 4, 5, 6 Hamming distance: 6 5. Between 010101 and 110110: Positions at which bits differ: 2, 4 Hamming distance: 2 6. Between 101010 and 110110: Positions at which bits differ: 1, 3, 5 Hamming distance: 3 The minimum Hamming distance among the codewords is 2, which is the distance between codewords 010101 and 110110. |
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| deepseekr1 | The Hamming pairwise distances between the codewords are calculated as follows: 1. **000000 vs 010101**: 3 differences → Distance = 3 2. **000000 vs 101010**: 3 differences → Distance = 3 3. **000000 vs 110110**: 4 differences → Distance = 4 4. **010101 vs 101010**: All bits differ → Distance = 6 5. **010101 vs 110110**: 3 differences → Distance = 3 6. **101010 vs 110110**: 3 differences → Distance = 3 The smallest Hamming distance among all pairs is **3**. **Answer:** The minimum Hamming distance is \boxed{3}. |
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| gpto1 | **Solution:** We are given the following codewords: 1. **C₁:** 0 0 0 0 0 0 2. **C₂:** 0 1 0 1 0 1 3. **C₃:** 1 0 1 0 1 0 4. **C₄:** 1 1 0 1 1 0 Our goal is to calculate the Hamming distances between all pairs of codewords and determine the minimum Hamming distance. --- ### **Step 1: List all pairs of codewords** Since there are 4 codewords, there are \(\binom{4}{2} = 6\) unique pairs: 1. \( C₁ \) and \( C₂ \) 2. \( C₁ \) and \( C₃ \) 3. \( C₁ \) and \( C₄ \) 4. \( C₂ \) and \( C₃ \) 5. \( C₂ \) and \( C₄ \) 6. \( C₃ \) and \( C₄ \) --- ### **Step 2: Calculate the Hamming distance for each pair** The **Hamming distance** between two codewords is the number of positions at which the corresponding bits are different. --- #### **Pair 1: \( C₁ \) and \( C₂ \)** Comparing \( C₁: \) 0 0 0 0 0 0 with \( C₂: \) 0 1 0 1 0 1 | Position | \( C₁ \) | \( C₂ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 0 | No | | 2 | 0 | 1 | **Yes** | | 3 | 0 | 0 | No | | 4 | 0 | 1 | **Yes** | | 5 | 0 | 0 | No | | 6 | 0 | 1 | **Yes** | **Total differences:** 3 **Hamming distance between \( C₁ \) and \( C₂ \) is 3.** --- #### **Pair 2: \( C₁ \) and \( C₃ \)** Comparing \( C₁: \) 0 0 0 0 0 0 with \( C₃: \) 1 0 1 0 1 0 | Position | \( C₁ \) | \( C₃ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 1 | **Yes** | | 2 | 0 | 0 | No | | 3 | 0 | 1 | **Yes** | | 4 | 0 | 0 | No | | 5 | 0 | 1 | **Yes** | | 6 | 0 | 0 | No | **Total differences:** 3 **Hamming distance between \( C₁ \) and \( C₃ \) is 3.** --- #### **Pair 3: \( C₁ \) and \( C₄ \)** Comparing \( C₁: \) 0 0 0 0 0 0 with \( C₄: \) 1 1 0 1 1 0 | Position | \( C₁ \) | \( C₄ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 1 | **Yes** | | 2 | 0 | 1 | **Yes** | | 3 | 0 | 0 | No | | 4 | 0 | 1 | **Yes** | | 5 | 0 | 1 | **Yes** | | 6 | 0 | 0 | No | **Total differences:** 4 **Hamming distance between \( C₁ \) and \( C₄ \) is 4.** --- #### **Pair 4: \( C₂ \) and \( C₃ \)** Comparing \( C₂: \) 0 1 0 1 0 1 with \( C₃: \) 1 0 1 0 1 0 | Position | \( C₂ \) | \( C₃ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 1 | **Yes** | | 2 | 1 | 0 | **Yes** | | 3 | 0 | 1 | **Yes** | | 4 | 1 | 0 | **Yes** | | 5 | 0 | 1 | **Yes** | | 6 | 1 | 0 | **Yes** | **Total differences:** 6 **Hamming distance between \( C₂ \) and \( C₃ \) is 6.** --- #### **Pair 5: \( C₂ \) and \( C₄ \)** Comparing \( C₂: \) 0 1 0 1 0 1 with \( C₄: \) 1 1 0 1 1 0 | Position | \( C₂ \) | \( C₄ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 1 | **Yes** | | 2 | 1 | 1 | No | | 3 | 0 | 0 | No | | 4 | 1 | 1 | No | | 5 | 0 | 1 | **Yes** | | 6 | 1 | 0 | **Yes** | **Total differences:** 3 **Hamming distance between \( C₂ \) and \( C₄ \) is 3.** --- #### **Pair 6: \( C₃ \) and \( C₄ \)** Comparing \( C₃: \) 1 0 1 0 1 0 with \( C₄: \) 1 1 0 1 1 0 | Position | \( C₃ \) | \( C₄ \) | Different? | |----------|----------|----------|------------| | 1 | 1 | 1 | No | | 2 | 0 | 1 | **Yes** | | 3 | 1 | 0 | **Yes** | | 4 | 0 | 1 | **Yes** | | 5 | 1 | 1 | No | | 6 | 0 | 0 | No | **Total differences:** 3 **Hamming distance between \( C₃ \) and \( C₄ \) is 3.** --- ### **Step 3: Summary of Hamming distances** - **\( C₁ \) and \( C₂ \):** 3 - **\( C₁ \) and \( C₃ \):** 3 - **\( C₁ \) and \( C₄ \):** 4 - **\( C₂ \) and \( C₃ \):** 6 - **\( C₂ \) and \( C₄ \):** 3 - **\( C₃ \) and \( C₄ \):** 3 --- ### **Step 4: Determine the minimum Hamming distance** From the calculated distances, the minimum Hamming distance is: - **Minimum Hamming distance:** 3 --- **Answer:** The pairwise Hamming distances among the codewords are: - Between \( C₁ \) and \( C₂ \): **3** - Between \( C₁ \) and \( C₃ \): **3** - Between \( C₁ \) and \( C₄ \): **4** - Between \( C₂ \) and \( C₃ \): **6** - Between \( C₂ \) and \( C₄ \): **3** - Between \( C₃ \) and \( C₄ \): **3** Therefore, the **minimum Hamming distance** among the codewords is **3**. |
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| GPT-4o | To solve the problem of calculating the Hamming pairwise distances and determining the minimum Hamming distance among the codewords \(000000\), \(010101\), \(101010\), and \(110110\), we need to follow these steps: 1. **Calculate the Hamming distance between each pair of codewords:** The Hamming distance between two codewords is the number of positions at which the corresponding bits are different. 2. **List all the pairwise Hamming distances:** We have a total of \( \binom{4}{2} = 6 \) pairs to consider since there are 4 codewords. 3. **Identify the minimum Hamming distance among these pairs.** Let's calculate the Hamming distances: - **Between \(000000\) and \(010101\):** - Compare each bit: - 0 vs 0: same - 0 vs 1: different - 0 vs 0: same - 0 vs 1: different - 0 vs 0: same - 0 vs 1: different - Hamming distance = 3 - **Between \(000000\) and \(101010\):** - Compare each bit: - 0 vs 1: different - 0 vs 0: same - 0 vs 1: different - 0 vs 0: same - 0 vs 1: different - 0 vs 0: same - Hamming distance = 3 - **Between \(000000\) and \(110110\):** - Compare each bit: - 0 vs 1: different - 0 vs 1: different - 0 vs 0: same - 0 vs 1: different - 0 vs 1: different - 0 vs 0: same - Hamming distance = 4 - **Between \(010101\) and \(101010\):** - Compare each bit: - 0 vs 1: different - 1 vs 0: different - 0 vs 1: different - 1 vs 0: different - 0 vs 1: different - 1 vs 0: different - Hamming distance = 6 - **Between \(010101\) and \(110110\):** - Compare each bit: - 0 vs 1: different - 1 vs 1: same - 0 vs 0: same - 1 vs 1: same - 0 vs 1: different - 1 vs 0: different - Hamming distance = 3 - **Between \(101010\) and \(110110\):** - Compare each bit: - 1 vs 1: same - 0 vs 1: different - 1 vs 0: different - 0 vs 1: different - 1 vs 1: same - 0 vs 0: same - Hamming distance = 3 Now, let's list all the calculated Hamming distances: - 3 - 3 - 4 - 6 - 3 - 3 The minimum Hamming distance among these pairs is \(3\). **Solution:** The minimum Hamming distance among the codewords \(000000\), \(010101\), \(101010\), and \(110110\) is \(3\). |
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| llama3.1-70B-Instruct | To calculate the Hamming pairwise distances and determine the minimum Hamming distance among the given codewords, we need to compare each pair of codewords and count the number of positions at which they differ. The given codewords are: 1. 000000 2. 010101 3. 101010 4. 110110 Let's calculate the Hamming distances between each pair of codewords: 1. 000000 vs. 010101: 3 (positions 2, 4, and 6 differ) 2. 000000 vs. 101010: 3 (positions 1, 3, and 5 differ) 3. 000000 vs. 110110: 4 (positions 1, 2, 5, and 6 differ) 4. 010101 vs. 101010: 4 (positions 1, 2, 3, and 5 differ) 5. 010101 vs. 110110: 2 (positions 1 and 5 differ) 6. 101010 vs. 110110: 2 (positions 2 and 6 differ) The minimum Hamming distance among these codewords is 2, which occurs between the pairs (010101, 110110) and (101010, 110110). Therefore, the minimum Hamming distance among the given codewords is 2. |
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| llama3.1-405B-Instruct | To calculate the Hamming pairwise distances and determine the minimum Hamming distance among the given codewords, we can follow these steps: 1. Convert each codeword into a binary vector. 2. Calculate the Hamming distance between each pair of codewords. 3. Find the minimum Hamming distance among all pairs. Here are the calculations: **Step 1: Convert each codeword into a binary vector** * 000000 -> [0, 0, 0, 0, 0, 0] * 010101 -> [0, 1, 0, 1, 0, 1] * 101010 -> [1, 0, 1, 0, 1, 0] * 110110 -> [1, 1, 0, 1, 1, 0] **Step 2: Calculate the Hamming distance between each pair of codewords** * [0, 0, 0, 0, 0, 0] vs. [0, 1, 0, 1, 0, 1] = 3 (different bits: 2, 4, 6) * [0, 0, 0, 0, 0, 0] vs. [1, 0, 1, 0, 1, 0] = 3 (different bits: 1, 3, 5) * [0, 0, 0, 0, 0, 0] vs. [1, 1, 0, 1, 1, 0] = 4 (different bits: 1, 2, 4, 5) * [0, 1, 0, 1, 0, 1] vs. [1, 0, 1, 0, 1, 0] = 6 (different bits: 1, 2, 3, 4, 5, 6) * [0, 1, 0, 1, 0, 1] vs. [1, 1, 0, 1, 1, 0] = 3 (different bits: 1, 3, 5) * [1, 0, 1, 0, 1, 0] vs. [1, 1, 0, 1, 1, 0] = 3 (different bits: 2, 4, 6) **Step 3: Find the minimum Hamming distance among all pairs** The minimum Hamming distance is 3, which occurs between the following pairs: * [0, 0, 0, 0, 0, 0] vs. [0, 1, 0, 1, 0, 1] * [0, 0, 0, 0, 0, 0] vs. [1, 0, 1, 0, 1, 0] * [0, 1, 0, 1, 0, 1] vs. [1, 1, 0, 1, 1, 0] * [1, 0, 1, 0, 1, 0] vs. [1, 1, 0, 1, 1, 0] Therefore, the minimum Hamming distance among the given codewords is 3. |
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| DeepSeek-Coder-V2 | To solve the problem of calculating the Hamming pairwise distances and determining the minimum Hamming distance among the given codewords, we will follow these steps: 1. **Define the Hamming Distance**: The Hamming distance between two strings of equal length is the number of positions at which the corresponding bits are different. 2. **List the Codewords**: The given codewords are: - 000000 - 010101 - 101010 - 110110 3. **Calculate Pairwise Hamming Distances**: - **Distance between 000000 and 010101**: - Differences: 2nd, 4th, and 6th positions. - Distance = 3. - **Distance between 000000 and 101010**: - Differences: 1st, 3rd, and 5th positions. - Distance = 3. - **Distance between 000000 and 110110**: - Differences: 1st, 2nd, 4th, and 6th positions. - Distance = 4. - **Distance between 010101 and 101010**: - Differences: 1st, 2nd, 3rd, 4th, 5th, and 6th positions. - Distance = 6. - **Distance between 010101 and 110110**: - Differences: 1st, 3rd, and 5th positions. - Distance = 3. - **Distance between 101010 and 110110**: - Differences: 2nd, 4th, and 6th positions. - Distance = 3. 4. **Determine the Minimum Hamming Distance**: - The calculated distances are: 3, 3, 4, 6, 3, 3. - The minimum Hamming distance is 3. Thus, the minimum Hamming distance among the given codewords is $\boxed{3}$. |
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| Qwen2-72B-Instruct | To calculate the Hamming pairwise distances between the codewords, we need to compare each pair of codewords and count the number of positions at which the corresponding bits are different. The codewords are: 000000, 010101, 101010, 110110. Let's calculate the Hamming distances: 1. Between 000000 and 010101: Positions at which bits differ: 1, 2, 4, 5, 6 Hamming distance: 5 2. Between 000000 and 101010: Positions at which bits differ: 1, 3, 4, 6 Hamming distance: 4 3. Between 000000 and 110110: Positions at which bits differ: 1, 2, 3, 5, 6 Hamming distance: 5 4. Between 010101 and 101010: Positions at which bits differ: 1, 2, 3, 4, 5, 6 Hamming distance: 6 5. Between 010101 and 110110: Positions at which bits differ: 2, 4 Hamming distance: 2 6. Between 101010 and 110110: Positions at which bits differ: 1, 3, 5 Hamming distance: 3 The minimum Hamming distance among the codewords is 2, which is the distance between codewords 010101 and 110110. |
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| baichuan4 | To calculate the Hamming pairwise distances between the codewords 000000, 010101, 101010, and 110110, we need to compare each pair of codewords and count the number of positions at which the corresponding bits are different. Here are the pairwise Hamming distances: 1. Distance between 000000 and 010101: 3 (bits at positions 2, 4, and 6 are different) 2. Distance between 000000 and 101010: 3 (bits at positions 1, 3, and 5 are different) 3. Distance between 000000 and 110110: 4 (bits at positions 1, 2, 4, and 6 are different) 4. Distance between 010101 and 101010: 4 (bits at positions 1, 2, 5, and 6 are different) 5. Distance between 010101 and 110110: 3 (bits at positions 1, 3, and 5 are different) 6. Distance between 101010 and 110110: 2 (bits at positions 3 and 6 are different) Now, we need to determine the minimum Hamming distance among these pairwise distances. From the list above, the minimum Hamming distance is 2, which is the distance between the codewords 101010 and 110110. |
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| doubao | The Hamming distance between two codewords is the number of positions where they differ. The Hamming distance between 000000 and 010101 is 3. The Hamming distance between 000000 and 101010 is 3. The Hamming distance between 000000 and 110110 is 3. The Hamming distance between 010101 and 101010 is 3. The Hamming distance between 010101 and 110110 is 6. The Hamming distance between 101010 and 110110 is 3. The minimum Hamming distance is 3. |
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| minimax | To calculate the Hamming distance between two codewords, we count the number of positions at which the corresponding symbols are different. The Hamming distance between two codewords of the same length is the number of positions in which the corresponding symbols are different. Let's calculate the pairwise Hamming distances between the given codewords: 1. Hamming distance between 000000 and 010101: *** *** Hamming distance = 3 (positions 2, 4, and 6 are different) 2. Hamming distance between 000000 and 101010: *** *** Hamming distance = 3 (positions 1, 3, and 5 are different) 3. Hamming distance between 000000 and 110110: *** *** Hamming distance = 4 (positions 1, 2, 4, and 6 are different) 4. Hamming distance between 010101 and 101010: *** *** Hamming distance = 4 (positions 1, 2, 3, and 5 are different) 5. Hamming distance between 010101 and 110110: *** *** Hamming distance = 3 (positions 1, 3, and 6 are different) 6. Hamming distance between 101010 and 110110: *** *** Hamming distance = 3 (positions 2, 4, and 5 are different) Now, we determine the minimum Hamming distance among the calculated distances: - The minimum Hamming distance is 3. Therefore, the minimum Hamming distance among the given codewords is 3. |
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| kimi | To calculate the Hamming pairwise distances, we need to compare each pair of codewords and count the number of positions at which the corresponding symbols are different. Here are the given codewords: 1. 000000 2. 010101 3. 101010 4. 110110 Let's calculate the Hamming distance for each pair: 1. Hamming distance between 000000 and 010101: - Different positions: 1, 2, 3, 4, 5, 6 - Distance: 6 2. Hamming distance between 000000 and 101010: - Different positions: 1, 2, 3, 4, 5, 6 - Distance: 6 3. Hamming distance between 000000 and 110110: - Different positions: 1, 2, 3, 4, 5, 6 - Distance: 6 4. Hamming distance between 010101 and 101010: - Different positions: 1, 3, 5 - Distance: 3 5. Hamming distance between 010101 and 110110: - Different positions: 1, 2, 4, 5, 6 - Distance: 5 6. Hamming distance between 101010 and 110110: - Different positions: 2, 3, 4, 6 - Distance: 4 Now, let's determine the minimum Hamming distance among these distances: - The distances calculated are 6, 6, 6, 3, 5, and 4. The minimum Hamming distance among the given codewords is 3, which occurs between the codewords 010101 and 101010. |
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| wenxinyiyan | To calculate the Hamming pairwise distances and determine the minimum Hamming distance among the given codewords, we need to follow these steps: Step 1: Define the codewords. Let's denote the codewords as follows: C1 = 000000 C2 = 010101 C3 = 101010 C4 = 110110 Step 2: Calculate the Hamming distance between each pair of codewords. The Hamming distance between two codewords is the number of positions where the corresponding bits are different. * Distance between C1 and C2: C1: 000000 C2: 010101 Different bits: 3 (positions 2, 4, 6) Hamming distance (C1, C2) = 3 * Distance between C1 and C3: C1: 000000 C3: 101010 Different bits: 3 (positions 1, 3, 5) Hamming distance (C1, C3) = 3 * Distance between C1 and C4: C1: 000000 C4: 110110 Different bits: 4 (positions 1, 2, 4, 5) Hamming distance (C1, C4) = 4 * Distance between C2 and C3: C2: 010101 C3: 101010 Different bits: 6 (all positions) Hamming distance (C2, C3) = 6 * Distance between C2 and C4: C2: 010101 C4: 110110 Different bits: 3 (positions 1, 3, 6) Hamming distance (C2, C4) = 3 * Distance between C3 and C4: C3: 101010 C4: 110110 Different bits: 3 (positions 2, 3, 4) Hamming distance (C3, C4) = 3 Step 3: Determine the minimum Hamming distance. From the calculated pairwise distances, the minimum Hamming distance is 3. Therefore, the minimum Hamming distance among the given codewords is 3. |
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| zhipu | The Hamming distance between two codewords is the number of positions at which the corresponding symbols are different. In other words, it's the number of bit flips required to change one codeword into another. Given the codewords: 000000, 010101, 101010, and 110110, we will calculate the pairwise Hamming distances and then determine the minimum Hamming distance among them. Let's calculate the pairwise distances: 1. Distance between 000000 and 010101: - Compare each bit: 000000 010101 - Differences are at positions 2, 4, and 6. - Hamming distance = 3 2. Distance between 000000 and 101010: - Compare each bit: 000000 101010 - Differences are at positions 1, 3, and 5. - Hamming distance = 3 3. Distance between 000000 and 110110: - Compare each bit: 000000 110110 - Differences are at positions 1, 2, 5, and 6. - Hamming distance = 4 4. Distance between 010101 and 101010: - Compare each bit: 010101 101010 - Differences are at all positions. - Hamming distance = 6 5. Distance between 010101 and 110110: - Compare each bit: 010101 110110 - Differences are at positions 1, 3, 4, and 6. - Hamming distance = 4 6. Distance between 101010 and 110110: - Compare each bit: 101010 110110 - Differences are at positions 2 and 5. - Hamming distance = 2 Now we have all the pairwise Hamming distances: - 000000 and 010101: 3 - 000000 and 101010: 3 - 000000 and 110110: 4 - 010101 and 101010: 6 - 010101 and 110110: 4 - 101010 and 110110: 2 The minimum Hamming distance among these pairs is 2, which is the distance between the codewords 101010 and 110110. So, the minimum Hamming distance among the given codewords is 2. |
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| gpto1 | **Solution:** We are given the following codewords: 1. **C₁:** 0 0 0 0 0 0 2. **C₂:** 0 1 0 1 0 1 3. **C₃:** 1 0 1 0 1 0 4. **C₄:** 1 1 0 1 1 0 Our goal is to calculate the Hamming distances between all pairs of codewords and determine the minimum Hamming distance. --- ### **Step 1: List all pairs of codewords** Since there are 4 codewords, there are \(\binom{4}{2} = 6\) unique pairs: 1. \( C₁ \) and \( C₂ \) 2. \( C₁ \) and \( C₃ \) 3. \( C₁ \) and \( C₄ \) 4. \( C₂ \) and \( C₃ \) 5. \( C₂ \) and \( C₄ \) 6. \( C₃ \) and \( C₄ \) --- ### **Step 2: Calculate the Hamming distance for each pair** The **Hamming distance** between two codewords is the number of positions at which the corresponding bits are different. --- #### **Pair 1: \( C₁ \) and \( C₂ \)** Comparing \( C₁: \) 0 0 0 0 0 0 with \( C₂: \) 0 1 0 1 0 1 | Position | \( C₁ \) | \( C₂ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 0 | No | | 2 | 0 | 1 | **Yes** | | 3 | 0 | 0 | No | | 4 | 0 | 1 | **Yes** | | 5 | 0 | 0 | No | | 6 | 0 | 1 | **Yes** | **Total differences:** 3 **Hamming distance between \( C₁ \) and \( C₂ \) is 3.** --- #### **Pair 2: \( C₁ \) and \( C₃ \)** Comparing \( C₁: \) 0 0 0 0 0 0 with \( C₃: \) 1 0 1 0 1 0 | Position | \( C₁ \) | \( C₃ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 1 | **Yes** | | 2 | 0 | 0 | No | | 3 | 0 | 1 | **Yes** | | 4 | 0 | 0 | No | | 5 | 0 | 1 | **Yes** | | 6 | 0 | 0 | No | **Total differences:** 3 **Hamming distance between \( C₁ \) and \( C₃ \) is 3.** --- #### **Pair 3: \( C₁ \) and \( C₄ \)** Comparing \( C₁: \) 0 0 0 0 0 0 with \( C₄: \) 1 1 0 1 1 0 | Position | \( C₁ \) | \( C₄ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 1 | **Yes** | | 2 | 0 | 1 | **Yes** | | 3 | 0 | 0 | No | | 4 | 0 | 1 | **Yes** | | 5 | 0 | 1 | **Yes** | | 6 | 0 | 0 | No | **Total differences:** 4 **Hamming distance between \( C₁ \) and \( C₄ \) is 4.** --- #### **Pair 4: \( C₂ \) and \( C₃ \)** Comparing \( C₂: \) 0 1 0 1 0 1 with \( C₃: \) 1 0 1 0 1 0 | Position | \( C₂ \) | \( C₃ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 1 | **Yes** | | 2 | 1 | 0 | **Yes** | | 3 | 0 | 1 | **Yes** | | 4 | 1 | 0 | **Yes** | | 5 | 0 | 1 | **Yes** | | 6 | 1 | 0 | **Yes** | **Total differences:** 6 **Hamming distance between \( C₂ \) and \( C₃ \) is 6.** --- #### **Pair 5: \( C₂ \) and \( C₄ \)** Comparing \( C₂: \) 0 1 0 1 0 1 with \( C₄: \) 1 1 0 1 1 0 | Position | \( C₂ \) | \( C₄ \) | Different? | |----------|----------|----------|------------| | 1 | 0 | 1 | **Yes** | | 2 | 1 | 1 | No | | 3 | 0 | 0 | No | | 4 | 1 | 1 | No | | 5 | 0 | 1 | **Yes** | | 6 | 1 | 0 | **Yes** | **Total differences:** 3 **Hamming distance between \( C₂ \) and \( C₄ \) is 3.** --- #### **Pair 6: \( C₃ \) and \( C₄ \)** Comparing \( C₃: \) 1 0 1 0 1 0 with \( C₄: \) 1 1 0 1 1 0 | Position | \( C₃ \) | \( C₄ \) | Different? | |----------|----------|----------|------------| | 1 | 1 | 1 | No | | 2 | 0 | 1 | **Yes** | | 3 | 1 | 0 | **Yes** | | 4 | 0 | 1 | **Yes** | | 5 | 1 | 1 | No | | 6 | 0 | 0 | No | **Total differences:** 3 **Hamming distance between \( C₃ \) and \( C₄ \) is 3.** --- ### **Step 3: Summary of Hamming distances** - **\( C₁ \) and \( C₂ \):** 3 - **\( C₁ \) and \( C₃ \):** 3 - **\( C₁ \) and \( C₄ \):** 4 - **\( C₂ \) and \( C₃ \):** 6 - **\( C₂ \) and \( C₄ \):** 3 - **\( C₃ \) and \( C₄ \):** 3 --- ### **Step 4: Determine the minimum Hamming distance** From the calculated distances, the minimum Hamming distance is: - **Minimum Hamming distance:** 3 --- **Answer:** The pairwise Hamming distances among the codewords are: - Between \( C₁ \) and \( C₂ \): **3** - Between \( C₁ \) and \( C₃ \): **3** - Between \( C₁ \) and \( C₄ \): **4** - Between \( C₂ \) and \( C₃ \): **6** - Between \( C₂ \) and \( C₄ \): **3** - Between \( C₃ \) and \( C₄ \): **3** Therefore, the **minimum Hamming distance** among the codewords is **3**. |
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| deepseekv3 | server error: Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-1aaa566d-4f36-9848-8e8f-2f75912f6493', 'request_id': '1aaa566d-4f36-9848-8e8f-2f75912f6493'} | 3 |