BCH Codes and Reed-Solomon Codes

eric.jacobsen@delete.ieee.org (Eric Jacobsen) writes:

> On Tue, 30 Mar 2004 11:52:17 -0500, Raymond Toy <toy@rtp.ericsson.se>
> wrote:
>
>>>>>>> "Randy" == Randy Yates <randy.yates@sonyericsson.com> writes:
>>
>>    Randy> "A BCH code over GF(p)" is a contradiction according to the way it has
>>    Randy> been presented to me. A BCH code is, BY DEFINITION, a BINARY code, 
>>    Randy> according to my notes.
>>
>>I had learned that a Reed-Solomon code was a special type of
>>(non-binary) BCH code.  So I guess it depends on the book you are
>>using as your reference.  I always take BCH to be the non-binary one,
>>unless it's clear from context that some other BCH code is being
>>discussed. 
>>
>>Ray
>
> That's my understanding as well, RS codes are a subset of BCH, not the
> other way around.  I've seen this classification and description in
> multiple places, so I tend to think of them that way.  Makes sense to
> me as well.

Hi Eric/Ray, 

Could you please give me a reference or two on where RS codes are defined
this way?

RS codes were defined to us in terms of the Vandermonde matrix over GF(q) 
of dimension 2*t by n, where t is the number of correctable errors and 
2t <= q-1. Here q can be any p^m. Then it was narrowed to BCH codes by
the codewords that only have elements over GF(p).
-- 
%  Randy Yates                  % "Ticket to the moon, flight leaves here today 
%% Fuquay-Varina, NC            %  from Satellite 2"
%%% 919-577-9882                % 'Ticket To The Moon' 
%%%% <yates@ieee.org>           % *Time*, Electric Light Orchestra
http://home.earthlink.net/~yatescr

Randy Yates <yates@ieee.org> writes:
> [...]
> Then it was narrowed to BCH codes by
> the codewords that only have elements over GF(p).

I should have added "where p = 2". 
-- 
%  Randy Yates                  % "So now it's getting late,
%% Fuquay-Varina, NC            %    and those who hesitate
%%% 919-577-9882                %    got no one..."
%%%% <yates@ieee.org>           % 'Waterfall', *Face The Music*, ELO
http://home.earthlink.net/~yatescr

"Randy Yates" <yates@ieee.org> wrote in message
news:u105iohz.fsf@ieee.org...
> >>
> >>I had learned that a Reed-Solomon code was a special type of
> >>(non-binary) BCH code.  So I guess it depends on the book you are
> >>using as your reference.  I always take BCH to be the non-binary one,
> >>unless it's clear from context that some other BCH code is being
> >>discussed.
> >>
> >>Ray
> >
> > That's my understanding as well, RS codes are a subset of BCH, not the
> > other way around.  I've seen this classification and description in
> > multiple places, so I tend to think of them that way.  Makes sense to
> > me as well.
>
> Hi Eric/Ray,
>
> Could you please give me a reference or two on where RS codes are defined
> this way?

Hello Randy,

One classic that describes the RS codes as a special case of the BCH codes
is "Error Correcting Codes" by Peterson and Weldon, section 9.2. Another
reference that makes this same claim is "Coding and Information Theory" by
Roman, section 4.3.

I hope this helps.

-- 
Clay S. Turner, V.P.
Wireless Systems Engineering, Inc.
Satellite Beach, Florida 32937
(321) 777-7889
www.wse.biz
csturner@wse.biz





>
> RS codes were defined to us in terms of the Vandermonde matrix over GF(q)
> of dimension 2*t by n, where t is the number of correctable errors and
> 2t <= q-1. Here q can be any p^m. Then it was narrowed to BCH codes by
> the codewords that only have elements over GF(p).
> -- 
> %  Randy Yates                  % "Ticket to the moon, flight leaves here
today
> %% Fuquay-Varina, NC            %  from Satellite 2"
> %%% 919-577-9882                % 'Ticket To The Moon'
> %%%% <yates@ieee.org>           % *Time*, Electric Light Orchestra
> http://home.earthlink.net/~yatescr

>>>>> "Clay" == Clay S Turner <CSTurner@WSE.Biz> writes:

    Clay> Hello Randy,

    Clay> One classic that describes the RS codes as a special case of the BCH codes
    Clay> is "Error Correcting Codes" by Peterson and Weldon, section 9.2. Another
    Clay> reference that makes this same claim is "Coding and Information Theory" by
    Clay> Roman, section 4.3.

Add Michelson and Levesque to the list.  This is what I used to learn
about block codes.

Ray

On Wed, 31 Mar 2004, Raymond Toy wrote:

> Add Michelson and Levesque to the list.  This is what I used to learn
> about block codes.
>
> Ray
>

for algebraic block codes, i highly recommend MacWilliams and Sloane's
"the theory of error-correcting codes" as your second book.  it's a bit
dense, but if you are mathematically inclined, it's a very thorough
treatment of the subject matter.

this book defines Reed-Solomon codes as:

"A Reed-Solomon code over GF(q) is a BCH code of length N=q-1.  Of course
q is never 2.  Thus the length is the number of nonzero elements in the
ground field."

cyclic codes tend to work better over larger finite fields;  in fact, if i
recall correctly, this is one motivation for Forney's thesis on
concatenated codes:  if we wish to use these powerful cyclic codes, how do
we map it to the channel?  of course, this is among other things
contributed in the said thesis.

julius

-- 
The most rigorous proofs will be shown by vigorous handwaving.
http://www.mit.edu/~kusuma

opinion of author is not necessarily of the institute

Julius, Ray, Clay,

Thank you.

--Randy

Julius Kusuma <kusuma@mit.edu> writes:

> On Wed, 31 Mar 2004, Raymond Toy wrote:
> 
> > Add Michelson and Levesque to the list.  This is what I used to learn
> > about block codes.
> >
> > Ray
> >
> 
> for algebraic block codes, i highly recommend MacWilliams and Sloane's
> "the theory of error-correcting codes" as your second book.  it's a bit
> dense, but if you are mathematically inclined, it's a very thorough
> treatment of the subject matter.
> 
> this book defines Reed-Solomon codes as:
> 
> "A Reed-Solomon code over GF(q) is a BCH code of length N=q-1.  Of course
> q is never 2.  Thus the length is the number of nonzero elements in the
> ground field."
> 
> cyclic codes tend to work better over larger finite fields;  in fact, if i
> recall correctly, this is one motivation for Forney's thesis on
> concatenated codes:  if we wish to use these powerful cyclic codes, how do
> we map it to the channel?  of course, this is among other things
> contributed in the said thesis.
> 
> julius
> 
> -- 
> The most rigorous proofs will be shown by vigorous handwaving.
> http://www.mit.edu/~kusuma
> 
> opinion of author is not necessarily of the institute

-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124