In the book Information theory, Inference and Learning Algorithm, in chapter 13, MacKay defines bounded distance decoding
A bounded distance decoder is a decoder that returns the closest codeword to a received binary vector $\mathbf{r}$ if the distance from $\mathbf{r}$ to that codeword is less than or equal to $t$; otherwise it returns a failure message.
Could anybody provide an example of a simple linear code that can be decoded by such a bounded distance decoder?
The simplest example is a repetition code. You encode the bit 0 as 000 and the bit 1 as 111. The decoder outputs the majority bit, which is always at distance at most $t = 1$.
As another example, suppose we encode 0 as 0000 and 1 as 1111. We can still correct up to one error (by taking the majority), but we cannot accommodate $t = 2$, since given 0011 we can't tell whether it came from 0000 or from 1111. Therefore we need to declare failure.
Any perfect code will do. A perfect code is a $t-$error correcting code where the spheres of Hamming radius $t$ around the codewords cover the whole space.
Edit: In response to OP's comment, if the $t-$spheres around codewords overlap, the code, by definition, is not a $t-$error correcting code.
In general, all decoding algorithms used in practice are bounded distance decoding algorithms, since except for very short lengths, it is not feasible to do a brute force comparison of the received word with all codewords.
The Syndrome decoding of a linear code is one such example, see any coding textbook, as well as Wikipedia entry here
