# Regarding Generator Matrices of MDS codes?

Started by July 19, 2006
```Hi There,

I am a bit confused.  If I recall correctly, I have read/learned that
any submatrix of a generator matrix is invertible.  If one considers
the non-systematic Reed-Solomon code generated by a Vandermonde matrix,
this is true.

However, if one considers the "Reed-Solomon" code generated by [I | C],
where I is a k X k identity matrix, and C is a (k X n-k) Cauchy matrix,
not every submatrix is invertible.  However, every k X k submatrix will
be invertible.

Could someone clarify?

Also, the second code that I discussed, is that a Reed-Solomon code?
In literature I see that some people regard it as a Reed-Solomon code.
The two definitions that I know of for Reed-Solomon codes are either
the Vandermonde generator matrix (Reed and Solomon's original work) or
the n-k consecutive roots generator polynomial?

Your time and effort will greatly be appreciated
Jaco

```
```<jaco.versfeld@gmail.com> wrote in message
>  If I recall correctly, I have read/learned that
> any submatrix of a generator matrix is invertible.

It is not true in general that any submatrix of a generator matrix
is invertible.  The statement is not true for Reed-Solomon (more
generally, MDS) codes unless the assertion includes the further
qualification that the submatrix is of size k x k.  So, either your
recollection is faulty or you read/learned something that is incorrect.

> However, if one considers the "Reed-Solomon" code generated by [I | C],
> where I is a k X k identity matrix, and C is a (k X n-k) Cauchy matrix,
> not every submatrix is invertible.  However, every k X k submatrix will
> be invertible.

>Could someone clarify?

It is an axiom of the (much maligned in the U.S.) scientific method
that a theory that does not fit the facts must be discarded or modified.
You have a fact here that is counter to your theory that every submatrix
of a generator matrix is invertible.  What do you want to do with your
theory?

```
```Thank you very much.

Kind Regards,
Jaco

```