| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To calculate the Hamming distance between two codewords, we count the number of positions at which the corresponding bits are different. Let's calculate the Hamming distances between all pairs of the given codewords: 1. Between 00000 and 10101: Positions where bits are different: 1, 3, 5 Hamming distance: 3 2. Between 00000 and 01010: Positions where bits are different: 2, 4 Hamming distance: 2 3. Between 10101 and 01010: Positions where bits are different: 1, 2, 3, 4, 5 Hamming distance: 5 Now, we determine the minimum Hamming distance among these distances, which is 2. Therefore, the minimum Hamming distance among the given codewords is 2. |
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| deepseekr1 | To calculate the Hamming pairwise distances and determine the minimum among the codewords **00000**, **10101**, and **01010**, follow these steps: 1. **Compute pairwise Hamming distances**: - **Between 00000 and 10101**: Compare each bit: `0 vs 1` (diff), `0 vs 0`, `0 vs 1` (diff), `0 vs 0`, `0 vs 1` (diff). Total differences: **3** → Hamming distance = **3**. - **Between 00000 and 01010**: Compare each bit: `0 vs 0`, `0 vs 1` (diff), `0 vs 0`, `0 vs 1` (diff), `0 vs 0`. Total differences: **2** → Hamming distance = **2**. - **Between 10101 and 01010**: Compare each bit: `1 vs 0` (diff), `0 vs 1` (diff), `1 vs 0` (diff), `0 vs 1` (diff), `1 vs 0` (diff). Total differences: **5** → Hamming distance = **5**. 2. **Determine the minimum Hamming distance**: The pairwise distances are **3**, **2**, and **5**. The smallest value is **2**. **Final Answer**: The minimum Hamming distance among the codewords is \(\boxed{2}\). |
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| gpto1 | **Solution:** We are given three binary codewords of length 5: 1. \( C_1 = 0\ 0\ 0\ 0\ 0 \) 2. \( C_2 = 1\ 0\ 1\ 0\ 1 \) 3. \( C_3 = 0\ 1\ 0\ 1\ 0 \) Our task is to compute the Hamming distances between each pair of codewords and find the minimum Hamming distance among them. **Definition:** The **Hamming distance** between two codewords is the number of positions at which the corresponding symbols are different. Let's compute the pairwise Hamming distances step by step. --- ### 1. Hamming Distance between \( C_1 \) and \( C_2 \): **Compare each position:** | Position | \( C_1 \) | \( C_2 \) | Same/Different | |----------|-----------|-----------|----------------| | 1 | 0 | 1 | Different | | 2 | 0 | 0 | Same | | 3 | 0 | 1 | Different | | 4 | 0 | 0 | Same | | 5 | 0 | 1 | Different | **Total differences:** 3 So, the Hamming distance \( d(C_1, C_2) = 3 \). --- ### 2. Hamming Distance between \( C_1 \) and \( C_3 \): **Compare each position:** | Position | \( C_1 \) | \( C_3 \) | Same/Different | |----------|-----------|-----------|----------------| | 1 | 0 | 0 | Same | | 2 | 0 | 1 | Different | | 3 | 0 | 0 | Same | | 4 | 0 | 1 | Different | | 5 | 0 | 0 | Same | **Total differences:** 2 So, the Hamming distance \( d(C_1, C_3) = 2 \). --- ### 3. Hamming Distance between \( C_2 \) and \( C_3 \): **Compare each position:** | Position | \( C_2 \) | \( C_3 \) | Same/Different | |----------|-----------|-----------|----------------| | 1 | 1 | 0 | Different | | 2 | 0 | 1 | Different | | 3 | 1 | 0 | Different | | 4 | 0 | 1 | Different | | 5 | 1 | 0 | Different | **Total differences:** 5 So, the Hamming distance \( d(C_2, C_3) = 5 \). --- **Summary of Hamming Distances:** - \( d(C_1, C_2) = 3 \) - \( d(C_1, C_3) = 2 \) - \( d(C_2, C_3) = 5 \) **Minimum Hamming Distance:** The smallest Hamming distance among these codewords is **2**. --- **Answer:** The pairwise Hamming distances are: - Between **00000** and **10101**: distance = **3** - Between **00000** and **01010**: distance = **2** - Between **10101** and **01010**: distance = **5** **The minimum Hamming distance among the codewords is 2.** |
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| GPT-4o | To calculate the Hamming pairwise distances between the given codewords (00000, 10101, 01010) and determine the minimum Hamming distance, follow these steps: 1. **Hamming Distance Definition**: The Hamming distance between two binary strings of equal length is the number of positions at which the corresponding bits are different. 2. **Pairwise Comparisons**: Compare each pair of codewords to find the Hamming distances. ### Step-by-Step Calculation: 1. **Compare 00000 and 10101**: ``` 00000 10101 ----- 1 0 1 0 1 (differences at positions 1, 3, 5) ``` Hamming distance = 3 2. **Compare 00000 and 01010**: ``` 00000 01010 ----- 0 1 0 1 0 (differences at positions 2, 4) ``` Hamming distance = 2 3. **Compare 10101 and 01010**: ``` 10101 01010 ----- 1 1 1 1 1 (differences at all positions) ``` Hamming distance = 5 ### Summary of Hamming Distances: - Hamming distance between 00000 and 10101: 3 - Hamming distance between 00000 and 01010: 2 - Hamming distance between 10101 and 01010: 5 ### Determine the Minimum Hamming Distance: Among the calculated Hamming distances (3, 2, 5), the minimum Hamming distance is: \[ \boxed{2} \] Thus, the minimum Hamming distance among the codewords 00000, 10101, and 01010 is 2. |
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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. 00000 2. 10101 3. 01010 Let's calculate the Hamming distances between each pair of codewords: **Pair 1: 00000 and 10101** | Position | Codeword 1 | Codeword 2 | Difference | | --- | --- | --- | --- | | 1 | 0 | 1 | 1 | | 2 | 0 | 0 | 0 | | 3 | 0 | 1 | 1 | | 4 | 0 | 0 | 0 | | 5 | 0 | 1 | 1 | Hamming distance = 3 **Pair 2: 00000 and 01010** | Position | Codeword 1 | Codeword 2 | Difference | | --- | --- | --- | --- | | 1 | 0 | 0 | 0 | | 2 | 0 | 1 | 1 | | 3 | 0 | 0 | 0 | | 4 | 0 | 1 | 1 | | 5 | 0 | 0 | 0 | Hamming distance = 2 **Pair 3: 10101 and 01010** | Position | Codeword 1 | Codeword 2 | Difference | | --- | --- | --- | --- | | 1 | 1 | 0 | 1 | | 2 | 0 | 1 | 1 | | 3 | 1 | 0 | 1 | | 4 | 0 | 1 | 1 | | 5 | 1 | 0 | 1 | Hamming distance = 5 Now, let's find the minimum Hamming distance among these pairs: Minimum Hamming distance = 2 (between codewords 00000 and 01010) 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 need to compare each pair of codewords and count the number of positions at which they differ. Here are the given codewords: 1. 00000 2. 10101 3. 01010 Let's calculate the Hamming distance between each pair of codewords: 1. 00000 and 10101: * Position 1: 0 vs. 1 (different) * Position 2: 0 vs. 0 (same) * Position 3: 0 vs. 1 (different) * Position 4: 0 vs. 0 (same) * Position 5: 0 vs. 1 (different) Hamming distance: 3 2. 00000 and 01010: * Position 1: 0 vs. 0 (same) * Position 2: 0 vs. 1 (different) * Position 3: 0 vs. 0 (same) * Position 4: 0 vs. 1 (different) * Position 5: 0 vs. 0 (same) Hamming distance: 2 3. 10101 and 01010: * Position 1: 1 vs. 0 (different) * Position 2: 0 vs. 1 (different) * Position 3: 1 vs. 0 (different) * Position 4: 0 vs. 1 (different) * Position 5: 1 vs. 0 (different) Hamming distance: 5 Now, let's determine the minimum Hamming distance among the codewords: * The minimum Hamming distance between 00000 and 10101 is 3. * The minimum Hamming distance between 00000 and 01010 is 2. * The minimum Hamming distance between 10101 and 01010 is 5. Therefore, the minimum Hamming distance among the given codewords is 2, which occurs between 00000 and 01010. |
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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 codewords is the number of positions at which the corresponding bits are different. 2. **List the Codewords**: The given codewords are: - 00000 - 10101 - 01010 3. **Calculate the Hamming Distances**: - **Between 00000 and 10101**: - Compare each bit: - 0 vs 1 (different) - 0 vs 0 (same) - 0 vs 1 (different) - 0 vs 0 (same) - 0 vs 1 (different) - Total differences: 3 - Hamming distance: 3 - **Between 00000 and 01010**: - Compare each bit: - 0 vs 0 (same) - 0 vs 1 (different) - 0 vs 0 (same) - 0 vs 1 (different) - 0 vs 0 (same) - Total differences: 2 - Hamming distance: 2 - **Between 10101 and 01010**: - Compare each bit: - 1 vs 0 (different) - 0 vs 1 (different) - 1 vs 0 (different) - 0 vs 1 (different) - 1 vs 0 (different) - Total differences: 5 - Hamming distance: 5 4. **Determine the Minimum Hamming Distance**: - The Hamming distances calculated are: - 00000 to 10101: 3 - 00000 to 01010: 2 - 10101 to 01010: 5 - The minimum Hamming distance among these is 2. Therefore, the minimum Hamming distance among the given codewords is $\boxed{2}$. |
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| Qwen2-72B-Instruct | To calculate the Hamming distance between two codewords, we count the number of positions at which the corresponding bits are different. Let's calculate the Hamming distances between all pairs of the given codewords: 1. Between 00000 and 10101: Positions where bits are different: 1, 3, 5 Hamming distance: 3 2. Between 00000 and 01010: Positions where bits are different: 2, 4 Hamming distance: 2 3. Between 10101 and 01010: Positions where bits are different: 1, 2, 3, 4, 5 Hamming distance: 5 Now, we determine the minimum Hamming distance among these distances, which is 2. Therefore, the minimum Hamming distance among the given codewords is 2. |
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| baichuan4 | To calculate the Hamming pairwise distances between the codewords 00000, 10101, and 01010, we compare each pair of codewords and count the number of positions in which the corresponding bits are different. 1. Distance between 00000 and 10101: 00000 10101 ^ | 3 differences Hamming distance = 3 2. Distance between 00000 and 01010: 00000 01010 ^ | 2 differences Hamming distance = 2 3. Distance between 10101 and 01010: 10101 01010 ^ ^ | | 2 differences Hamming distance = 2 Now, we have calculated all pairwise Hamming distances. The minimum Hamming distance among these codewords is 2, which is the smallest value obtained from the pairwise comparisons. |
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| doubao | The Hamming distance between 00000 and 10101 is 5. The Hamming distance between 00000 and 01010 is 3. The Hamming distance between 10101 and 01010 is 5. The minimum Hamming distance is 3. |
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| minimax | To calculate the Hamming distance between two codewords, we compare the corresponding bits of the codewords and count the number of positions at which the bits are different. Let's calculate the pairwise Hamming distances between the given codewords: 1. Hamming distance between 00000 and 10101: *** *** Differences: 1 0 1 0 1 Hamming distance: 3 2. Hamming distance between 00000 and 01010: *** *** Differences: 0 1 0 1 0 Hamming distance: 2 3. Hamming distance between 10101 and 01010: *** *** Differences: 1 1 1 1 1 Hamming distance: 4 Now that we have the pairwise Hamming distances, we can determine the minimum Hamming distance among them: Minimum Hamming distance = min(3, 2, 4) = 2 Therefore, the minimum Hamming distance among the given codewords is 2. |
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| kimi | To calculate the Hamming pairwise distances between the given codewords, we need to compare each pair of codewords and count the number of positions at which the corresponding symbols are different. Here are the codewords: 1. 00000 2. 10101 3. 01010 Let's calculate the Hamming distances between each pair: 1. Distance between 00000 and 10101: - The first bit is different (0 vs 1) - The second bit is different (0 vs 1) - The third bit is the same (0 vs 0) - The fourth bit is different (0 vs 1) - The fifth bit is different (0 vs 1) - Total differences: 4 2. Distance between 00000 and 01010: - The first bit is the same (0 vs 0) - The second bit is different (0 vs 1) - The third bit is different (0 vs 1) - The fourth bit is the same (0 vs 0) - The fifth bit is different (0 vs 1) - Total differences: 3 3. Distance between 10101 and 01010: - The first bit is different (1 vs 0) - The second bit is the same (1 vs 1) - The third bit is different (0 vs 1) - The fourth bit is different (1 vs 0) - The fifth bit is different (1 vs 0) - Total differences: 3 Now, we have the distances: - Distance between 00000 and 10101: 4 - Distance between 00000 and 01010: 3 - Distance between 10101 and 01010: 3 The minimum Hamming distance among these codewords is 3, which is the distance between the pairs (00000, 01010) and (10101, 01010). |
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| wenxinyiyan | To calculate the Hamming pairwise distances between the given codewords (00000, 10101, 01010), we need to compare each pair of codewords and count the number of positions where the corresponding bits are different. Let's denote the codewords as follows: C1 = 00000 C2 = 10101 C3 = 01010 Now, we will calculate the Hamming distances between each pair: 1. Hamming distance between C1 and C2: Compare the corresponding bits of C1 and C2: C1: 0 0 0 0 0 C2: 1 0 1 0 1 The bits that are different are at positions 1, 3, and 5. So, the Hamming distance between C1 and C2 is 3. 2. Hamming distance between C1 and C3: Compare the corresponding bits of C1 and C3: C1: 0 0 0 0 0 C3: 0 1 0 1 0 The bits that are different are at positions 2, 4, and 5. So, the Hamming distance between C1 and C3 is 3. 3. Hamming distance between C2 and C3: Compare the corresponding bits of C2 and C3: C2: 1 0 1 0 1 C3: 0 1 0 1 0 The bits that are different are at positions 1, 2, 3, 4, and 5. So, the Hamming distance between C2 and C3 is 5. Now, to determine the minimum Hamming distance among the codewords, we take the minimum value from the calculated distances: Minimum Hamming distance = min(3, 3, 5) = 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. To calculate the pairwise Hamming distances among the given codewords (00000, 10101, 01010), we compare each pair of codewords and count the number of differences. Let's calculate the Hamming distances step by step: 1. Hamming distance between 00000 and 10101: - Compare each digit: - 0 vs 1 (different) - 0 vs 0 (same) - 0 vs 1 (different) - 0 vs 0 (same) - 0 vs 1 (different) - Total differences: 3 2. Hamming distance between 00000 and 01010: - Compare each digit: - 0 vs 0 (same) - 0 vs 1 (different) - 0 vs 0 (same) - 0 vs 1 (different) - 0 vs 0 (same) - Total differences: 2 3. Hamming distance between 10101 and 01010: - Compare each digit: - 1 vs 0 (different) - 0 vs 1 (different) - 1 vs 0 (different) - 0 vs 1 (different) - 1 vs 0 (different) - Total differences: 5 Now, we have the pairwise Hamming distances: - Between 00000 and 10101: 3 - Between 00000 and 01010: 2 - Between 10101 and 01010: 5 The minimum Hamming distance among these pairs is 2 (between 00000 and 01010). ### Summary: - Pairwise Hamming distances: - 00000 and 10101: 3 - 00000 and 01010: 2 - 10101 and 01010: 5 - Minimum Hamming distance: 2 Thus, the minimum Hamming distance among the given codewords is 2. |
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| gpto1 | **Solution:** We are given three binary codewords of length 5: 1. \( C_1 = 0\ 0\ 0\ 0\ 0 \) 2. \( C_2 = 1\ 0\ 1\ 0\ 1 \) 3. \( C_3 = 0\ 1\ 0\ 1\ 0 \) Our task is to compute the Hamming distances between each pair of codewords and find the minimum Hamming distance among them. **Definition:** The **Hamming distance** between two codewords is the number of positions at which the corresponding symbols are different. Let's compute the pairwise Hamming distances step by step. --- ### 1. Hamming Distance between \( C_1 \) and \( C_2 \): **Compare each position:** | Position | \( C_1 \) | \( C_2 \) | Same/Different | |----------|-----------|-----------|----------------| | 1 | 0 | 1 | Different | | 2 | 0 | 0 | Same | | 3 | 0 | 1 | Different | | 4 | 0 | 0 | Same | | 5 | 0 | 1 | Different | **Total differences:** 3 So, the Hamming distance \( d(C_1, C_2) = 3 \). --- ### 2. Hamming Distance between \( C_1 \) and \( C_3 \): **Compare each position:** | Position | \( C_1 \) | \( C_3 \) | Same/Different | |----------|-----------|-----------|----------------| | 1 | 0 | 0 | Same | | 2 | 0 | 1 | Different | | 3 | 0 | 0 | Same | | 4 | 0 | 1 | Different | | 5 | 0 | 0 | Same | **Total differences:** 2 So, the Hamming distance \( d(C_1, C_3) = 2 \). --- ### 3. Hamming Distance between \( C_2 \) and \( C_3 \): **Compare each position:** | Position | \( C_2 \) | \( C_3 \) | Same/Different | |----------|-----------|-----------|----------------| | 1 | 1 | 0 | Different | | 2 | 0 | 1 | Different | | 3 | 1 | 0 | Different | | 4 | 0 | 1 | Different | | 5 | 1 | 0 | Different | **Total differences:** 5 So, the Hamming distance \( d(C_2, C_3) = 5 \). --- **Summary of Hamming Distances:** - \( d(C_1, C_2) = 3 \) - \( d(C_1, C_3) = 2 \) - \( d(C_2, C_3) = 5 \) **Minimum Hamming Distance:** The smallest Hamming distance among these codewords is **2**. --- **Answer:** The pairwise Hamming distances are: - Between **00000** and **10101**: distance = **3** - Between **00000** and **01010**: distance = **2** - Between **10101** and **01010**: distance = **5** **The minimum Hamming distance among the codewords is 2.** |
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| deepseekv3 | To calculate the Hamming pairwise distances and determine the minimum Hamming distance among the codewords **00000**, **10101**, and **01010**, follow these steps: --- ### Step 1: Calculate the Hamming pairwise distances The **Hamming distance** between two codewords is the number of positions at which the corresponding bits differ. 1. **Distance between 00000 and 10101**: - Compare bit by bit: 0 vs 1 → differ 0 vs 0 → same 0 vs 1 → differ 0 vs 0 → same 0 vs 1 → differ - Total differences: **3** 2. **Distance between 00000 and 01010**: - Compare bit by bit: 0 vs 0 → same 0 vs 1 → differ 0 vs 0 → same 0 vs 1 → differ 0 vs 0 → same - Total differences: **2** 3. **Distance between 10101 and 01010**: - Compare bit by bit: 1 vs 0 → differ 0 vs 1 → differ 1 vs 0 → differ 0 vs 1 → differ 1 vs 0 → differ - Total differences: **5** --- ### Step 2: Determine the minimum Hamming distance The Hamming distances are **3**, **2**, and **5**. The **minimum Hamming distance** is the smallest of these values: **Minimum Hamming distance = 2** --- ### Final Answer: The minimum Hamming distance among the codewords **00000**, **10101**, and **01010** is **2**. |
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