Find $2^{2006}$ positive integers satisfying the following conditions.
(i) Each positive integer has $2^{2005}$ digits.
(ii) Each positive integer only has 7 or 8 in its digits.
(iii) Among any two chosen integers, at most half of their corresponding digits are the same.
1. **Define the sets and their properties:**
   - Let \( S_1 = \{77, 78\} \).
   - Define the inverse of an element such that 7 maps to 8 and 8 maps to 7. Thus, the inverse of \( S_1 \) is \( S_1' = \{88, 87\} \).
   - Define \( S_i^2 \) as the set obtained by concatenating each element of \( S_i \) with itself. For example, \( S_1^2 = \{7777, 7878\} \).
   - Define \( S_i + S_i' \) as the set obtained by concatenating each element of \( S_i \) with its corresponding element in \( S_i' \). For example, \( S_1 + S_1' = \{7788, 7887\} \).
   - Define \( T_i = S_i \cup S_i' \).
   - Define \( S_{i+1} = S_i^2 \cup (S_i + S_i') \).

2. **Claim:**
   - The set \( T_{2005} \) is the required set of \( 2^{2006} \) positive integers.

3. **Verification of conditions:**
   - Each number in \( T_{2005} \) has \( 2^{2005} \) digits.
   - Each number in \( T_{2005} \) only contains the digits 7 or 8.

4. **Induction to prove condition (iii):**
   - **Base Case:**
     - For \( T_1 \), the set \( \{77, 78, 88, 87\} \) clearly satisfies condition (iii) since any two numbers differ in at least one digit out of two.
   - **Inductive Step:**
     - Assume \( T_k \) satisfies condition (iii). We need to show \( T_{k+1} \) also satisfies condition (iii).
     - Consider \( T_{k+1} = S_k^2 \cup (S_k + S_k') \).
     - For any two elements in \( S_k^2 \), the first half and the second half of the digits are the same as in \( S_k \), so they differ in at most half of the digits.
     - For any two elements in \( S_k + S_k' \), the first half and the second half of the digits are the same as in \( S_k \) and \( S_k' \), so they differ in at most half of the digits.
     - Therefore, \( T_{k+1} \) satisfies condition (iii).

5. **Conclusion:**
   - By induction, \( T_{2005} \) satisfies all the given conditions.

The final answer is \( \boxed{ T_{2005} } \).