# Minimum Distance of Linear Binary Code

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.

• 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. Apr 2 '18 at 15:05
• @ChrisLeary Thank you for the insight! Apr 2 '18 at 15:08
• My pleasure. Glad to be of assistance. Apr 3 '18 at 2:07

1. The distance is based off the number of differences in entries, not just $1$'s.
• 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. Apr 2 '18 at 16:43