How Can I Devise a Code for Digits 0-9 with a Hamming Distance of 2?

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To devise a code for the digits 0 to 9 with a Hamming distance of 2, it is essential to represent each digit in binary. The challenge lies in ensuring that any two valid codewords differ by at least two bits. Initial attempts to find a single code that maintains this distance across all digits were unsuccessful, particularly when transitioning from binary representations of 0 to 7 to 8 and 9. It was noted that achieving a Hamming distance of 2 requires three redundancy bits, which means a total of 7 bits is necessary to encode the digits 0 through 9. This structure allows for the correction of single-bit errors, highlighting the importance of redundancy in error detection and correction schemes. Further exploration and refinement of the code are needed to ensure it meets the specified criteria for all digits.
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*Devise a code for the digits 0 to 9 whose Hamming distance is 2.*

My efforts to answer this problem are kind of hard to explain, but I'll try. First I wrote out the digits 0 to 9 in binary. Then I tried to find a number that was only 2 numbers different from each one (get 2 ones when XOR them), but there was no single code that worked for all 9 numbers. I found one that worked from 0 to 7, but once the digits changed to 1000 it didn't work anymore.

Am I even approaching this right? Please help.
 
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You need 3 redunancy bits for a distance of 2, which allows you to correct single bit errors. Since it takes 4 bits to represent the numbers 0 through 9, you need a 7 bit code.
 

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