Reed-Solomon error correction capacity

Raymond Toy wrote:


> Yes, I understand the RS code could produce no error indication, and
> the CRC could detect that.
> But the CRC takes extra bits, so if I added an extra parity word or
> two to the RS code, would I get better or worse performance than with
> the CRC?

This is a system question. What is the cost of an undetected random 
error in the data vs the cost of the whole data been rejected because of 
uncorrectable errors?

Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

Vladimir Vassilevsky  <antispam_bogus@hotmail.com> wrote:

>Raymond Toy wrote:

>> Yes, I understand the RS code could produce no error indication, and
>> the CRC could detect that.
>> But the CRC takes extra bits, so if I added an extra parity word or
>> two to the RS code, would I get better or worse performance than with
>> the CRC?

>This is a system question. 

Actually, it sounds like more of a mathematics question to me.
Rephrased, does a given number number of bits of CRC give
you better, worse, or about the same error-detection as the same 
number of added bits of RS redundancy?

Some would argue that the added RS redundancy is better, based
on a minimum weight argument.

Steve

On Sat, 8 Nov 2008 01:27:07 +0000 (UTC), spope33@speedymail.org (Steve
Pope) wrote:

>Vladimir Vassilevsky  <antispam_bogus@hotmail.com> wrote:
>
>>Raymond Toy wrote:
>
>>> Yes, I understand the RS code could produce no error indication, and
>>> the CRC could detect that.
>>> But the CRC takes extra bits, so if I added an extra parity word or
>>> two to the RS code, would I get better or worse performance than with
>>> the CRC?
>
>>This is a system question. 
>
>Actually, it sounds like more of a mathematics question to me.
>Rephrased, does a given number number of bits of CRC give
>you better, worse, or about the same error-detection as the same 
>number of added bits of RS redundancy?
>
>Some would argue that the added RS redundancy is better, based
>on a minimum weight argument.
>
>Steve

Could be, but in a system you can protect/cover multiple codewords
with a single CRC, so I think part of the answer may depend on the
nature of the transmissions.   If the transmissions are long enough, a
single CRC could protect a number of codewords and add relatively
little overhead.


Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.ericjacobsen.org

Blog: http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php

Steve Pope wrote:

> Vladimir Vassilevsky  <antispam_bogus@hotmail.com> wrote:
> 
> 
>>Raymond Toy wrote:
> 
> 
>>>Yes, I understand the RS code could produce no error indication, and
>>>the CRC could detect that.
>>>But the CRC takes extra bits, so if I added an extra parity word or
>>>two to the RS code, would I get better or worse performance than with
>>>the CRC?
> 
> 
>>This is a system question. 
> 
> 
> Actually, it sounds like more of a mathematics question to me.
> Rephrased, does a given number number of bits of CRC give
> you better, worse, or about the same error-detection as the same 
> number of added bits of RS redundancy?
> 
> Some would argue that the added RS redundancy is better, based
> on a minimum weight argument.

Assume the disastrous Eb/No with the huge amount of random errors.
Then the probability of the undetected error for RS code with K 
redundancy symbols in GF(M):

p = {sum C(M-1,x)}/M^K
       x <= K/2

You have to run the numbers to compare this to 1/2^N for CRC.


Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

Vladimir Vassilevsky  <antispam_bogus@hotmail.com> wrote:

>Assume the disastrous Eb/No with the huge amount of random errors.
>Then the probability of the undetected error for RS code with K 
>redundancy symbols in GF(M):

>p = {sum C(M-1,x)}/M^K
>       x <= K/2

I would say this formula is missing N, the total number
of symbols in a codeword.   I would replace the "M-1" with N.

I also think you need another factor of (M-1)^x within
the summation.

Steve

Thanks all for the information.