I'm working on an unassessed course problem (which I paraphrase for conciseness),
Let $C$ be the code over $\mathbb{F}_5$ with generator and parity-check matrices
$$G=\begin{pmatrix}2&3&4&1\end{pmatrix}, \hspace{1em} H=\begin{pmatrix}1&0&0&3\\2&1&0&3\\0&1&1&3\end{pmatrix}.$$
Determine the minimum distance $d(C)$ and the error-correcting index $t$,
and find the codeword of $C$ closest to $\underline{w}=1123$.
The solution booklet gives $$d=4, \hspace{1em} t=1, \hspace{1em} \underline{c}=1423 \text{ with } d(\underline{w},\underline{c})=1.$$
How can two words differ in fewer places than the minimum distance?
I'm guessing there's something basic I haven't understood.
Here is my intuitive way to see what is going on here.
Let us say , we have some way to put the Code Points in some linear order (which is not generally true : the "order" will be in some non-linear graph , but that is enough to see the intuition behind the Issue here)
Here the Integer Code Points are the Blue Marks , which may be something like $1423$.
The other Integer Points on the line are invalid Code Patterns. Eg the Purple Mark & the Green Mark , which may be like $1123$.
Over-all , the Minimum Distance is between 2 Blue Marks which is highlighted with the Arrows.
Now , we are given 2 invalid Integer Points , the Purple Mark & the Green Mark.
We want to know which valid Code Point got transmitted with errors , hence we chose the left Code Point for Purple Mark & the right Code Point for the Green Mark , because these are the closest valid Code Points to those invalid Points.
It might happen that the errors are too large & the transmitted Purple Point was actually the left-most & the transmitted Green Point was actually the right-most , though we will not know that : We will always select the nearest unique Blue Mark.
Now , we see that :
(1) the Distance between Valid Code Points is very large
(2) these Distances between Valid Code Points have a Minimum (larger is better)
(3) the Distance between valid Mark & Invalid Mark might be less (smaller is better)
(4) In fact , Distance between valid Mark & Invalid Mark SHOULD BE less than Minimum Distance between valid Code Points , to allow selecting the Correct Code Point when receiving invalid Core Points with errors
In text book Exercise , we received $1123$ which is invalid.
We calculate that the Original Code Point must have been $1423$ because that is the nearest , with Distance $1$.
Luckily , the error during transmission did not exceed $4$ , which is the Minimum Distance between 2 valid Code Points.
In Case , the error Distance had been 2 , we might have 2 or more valid Code Points to select (something like left-nearest , right-nearest , upper-nearest or lower-nearest) : we will not know which was the Original Code Point.