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I am new to understanding Coding Theory, and would not ask anybody just for the final answer, but rather the understanding/process.

Given an example question to calculate distance $(01010_2, 10101_2)$, I know the distance is based off the number of differences in $1$'s, equating to distance $= 5$.

However, how do I perform an operation to work out a "Minimum Distance" given a set $\{01010_2, 10101_2, 11011_2, 00100_2\}$? Would I do a similar operation as previously, but altogether at once?

Any illustration would greatly help me understand this.

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  • $\begingroup$ Add the two strings bitwise, mod 2. In your case you get the string 11111. The number of 1's in the "sum" tells you in how many places the strings disagree, hence the distance between them. If the strings agree in a place, you get 0+0 or 1+1, whose mod 2 sum is 0. $\endgroup$ Apr 2 '18 at 15:05
  • $\begingroup$ @ChrisLeary Thank you for the insight! $\endgroup$
    – Shivalkyr
    Apr 2 '18 at 15:08
  • $\begingroup$ My pleasure. Glad to be of assistance. $\endgroup$ Apr 3 '18 at 2:07
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  1. The distance is based off the number of differences in entries, not just $1$'s.
  2. For the minimum distance of a (small) given set: perform the distance operation to every distinct pair of words in the set; the minimum distance of your set will be the least number you compute.
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  • $\begingroup$ If I may confirm with you, based on the rule... given a set of ${00110011_2, 01101101_2, 01010110_2, 01010011_2}$, i got the minimum distance of 2, when comparing the difference between the first and the last in the set. $\endgroup$
    – Shivalkyr
    Apr 2 '18 at 16:43
  • $\begingroup$ @Shivalkyr Yes. Distance 2 occurred twice, didn't it? $\endgroup$
    – Rócherz
    Apr 2 '18 at 16:50
  • $\begingroup$ After checking it did occur twice, though are there any existing theorems that allow for a more efficient way of recognizing the Minimum Distance? Currently I just manually go through each distinct pair as you mentioned. $\endgroup$
    – Shivalkyr
    Apr 2 '18 at 16:51

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