My major is electrical engineering still undergraduate and I just enjoy learning math. I have started with a book of abstract algebra by Pinter. My question is In the remaining exercises, let $C$ be a code in, let $m$ denote the minimum distance in $C$, and let $a$ and $b$ denote codewords in $C$.
5) Prove that it is possible to detect up to $m − 1$ errors.
a) how can I find the minimum distance between two codewords when $a$ and $b$ are unknown ?
Note: The idea for me is clear, but still I cannot prove it mathematically.
If the minimum distance of the code is $m$ it means that there are pairs of codewords that differ on $m$ positions. There might be just one such pair or many more. But there are no pairs of codewords that differ in less than $m$ positions. So $d(C)=\min\{d(a,b)|a,b \in C,a\neq b\}$
An $m-1$ error is equivalent to receiving a codeword $u$ that differs on exactly $m-1$ positions from a codeword $a\in C$. Why is this error detectable? Because if $u\in C$ then $d(u,a)=m-1$ and then, the code's distance, $d(C)$, would be equal to $d(C)=m-1\lt m$.
What is perceived as a problem of a sufficiently rigorous mathematical proof arises from the fact that we suppose that we have an algorithm, a mechanism if you like that checks every received word by comparing it with all the codewords in $C$. But that supposition has to be made.