i wonder: how can i find minimal distance of a self correcting code in following situation: if we know that a code can fix every 3 errors(if not more than 3 errors, the word is recovered) and can detect every 5 errors(if between 3 and 5 errors, the algorithm will report that the error can not be fixed), how can we find its minimal distance?
i know that a code that fixes(hamming distance properties) $i$ number of errors costs a length of $2i+1$, and for detection of $i$ errors it costs a length of $i+1$. so the minimal length in this scenario is should been 7, but it cannot be 7 because distinguishing an error word E which can be obtained by discovering 5 errors or fixing 3 errors is undistinguisable in the following scenario: let a,b correct code words and E an erroronous word that can be obtained by 5 errors on the word x or by 3 errors from word y, so putting it like this makes it easier: x----E--Y, so it is not possible to distinguish between a word K that might have 3 errors or 5 errors in it, because its hamming distance from X or Y is the same. so, the minimal distance is 8.
what is the correct minimal hamming distance here?
**edit: please show me a formal and correct way to write the explanation why it cannot be 7 so i can learn correctly.
