<jaco.versfeld@gmail.com> wrote in message
news:1153314649.845377.152090@m79g2000cwm.googlegroups.com...
> 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?
Reply by ●July 19, 20062006-07-19
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