What is non-MDS codes?

Question

I am reading a paper where it is mentioned that $G=[I|P]$ is a systematic generator matrix of a linear non-MDS code C over a finite field. Here I am not very clear that what does "non-MDS code" means here. Kindly clarify if you can with an example. Thank you in advance. It will be a great help.

Answer 1

Well, by definition an MDS code is a linear code that achieves the singleton bound (so we have equality in the inequality).

So it stands to reason that a non-MDS code is one where the singleton bound is strict, so $$A_q(n,d) < q^{n-d+1}$$

So there must be some square submatrx of your $P$ that is singular, according to the characterisation theorem quoted on the linked page. So you can make your own example that way..