I am normally a speech/audio processing engineer, so this is somewhat out of my primary expertise area. I have recently implemented the Reed Solomon error correction algorithm and have gotten it to work for the following case: the roots of the generator polynomial are in positions 1 through 2t, where t is the number of errors. However, I implemented the case where the roots of the generator polynomial start at n, where n > 1. I looked through many library books and found a formula for finding the error values in only a couple of books. The problem is that it does not work. I found errors in several books concerning Reed Solomon, and there are a lot of details not explained in most texts. I am looking for reliable and accurate equations for finding the error values when the generator polynomial roots start in a position greater than 1. When I implemented the case where the generator polynomial roots start in position 1, you have the omega polynomial in the numerator and the derivative of the sigma polynomial in the denominator. The coefficient becomes 1. This works fine. The forumla for the case of n > 1, is the same except the coefficient is given as: - alpha^(j(1-n)) where n is the index for the first root of the generator polynomial. This doesn't work, as I am stress testing the algorithm over thousands of randomly generated messages. Thank you for any information, either formulas or books and articles or code that clearly explain this. I also have another question: we were asked to implement am (n,k) code, but the particular values of n and K are not listed in any table of valid BCH codes. The textbooks, however, refer to R-S as a type of BCH code. Does this mean that R-S cannot work with these values of n and h? I have a specification in front of me with these values of n and k, and also the AHA website seems to have an implementation of it running online (without the equations or source available). Is it possible to have an R-S decoder for values of (n,k) that are not listed in the tables of BCH codes? Linda Seltzer

# Reed Solomon error correction question

Started by ●October 7, 2008

Reply by ●October 7, 20082008-10-07

On Oct 7, 1:59�pm, "lindasel" <lselt...@alumni.caltech.edu> wrote:> I am looking for reliable and accurate equations for finding the error > values when the generator polynomial roots start in a position greater than > 1. � > Thank you for any information, either formulas or books and articles or > code that clearly explain this.I won't vouch for the reliability or accuracy of the information but you might want to look at the paper High-Speed Architectures for Reed-Solomon Decoders, IEEE Transactions on VLSI Systems, vol. 9, no. 5, Oct 2001 which is available (in Postscript format) at http://www.ifp.uiuc.edu/~sarwate/decoder.ps> I also have another question: we were asked to implement an (n,k) code, > but the particular values of n and k are not listed in any table of valid > BCH codes. �The textbooks, however, refer to R-S as a type of BCH code. > Does this mean that R-S cannot work with these values of n and k? �I have a > specification in front of me with these values of n and k, and also the AHA > website seems to have an implementation of it running online (without the > equations or source available). �Is it possible to have an R-S decoder for > values of (n,k) that are not listed in the tables of BCH codes?Yes, Reed-Solomon decoders can have values of n and k other than those listed in tables. The number of errors that can be corrected is (n-k)/2, and if n-k happens to be an odd number, then (n-k-1)/2 errors can be corrected and (n-k+1)/2 errors can be detected but not corrected. The values in published tables are usually for binary BCH codes, i.e. code symbols are bits, whereas the code symbols in Reed-Solomon codes are bytes (or bit vectors). --Dilip Sarwate

Reply by ●October 8, 20082008-10-08

Hi There,> > I am looking for reliable and accurate equations for finding the error > > values when the generator polynomial roots start in a position greater than > > 1. > > Thank you for any information, either formulas or books and articles or > > code that clearly explain this. > > I won't vouch for the reliability or accuracy of the information > but you might want to look at the paper High-Speed Architectures for > Reed-Solomon Decoders, IEEE Transactions on VLSI Systems, vol. 9, > no. 5, Oct 2001 which is available (in Postscript format) athttp://www.ifp.uiuc.edu/~sarwate/decoder.psI have implemented the algorithm in "http://www.ifp.uiuc.edu/~sarwate/ decoder.ps" using Matlab (a while ago, +/- 3 years), and according to my knowledge it is working great. I used it in several simulations. Thank you Prof Sarwate for sharing the link with us. Jaco _______________ Jaco Versfeld http://dept.ee.wits.ac.za/~versfeld "It is not the best team that wins, but the team that gets along best that wins" John C. Maxwell

Reply by ●October 9, 20082008-10-09

Thank you, Prof. Sarwate. Is there also a paper on the encoder? I have the following question about the encoder. In Bernard Sklar's paper on what communications engineers should know about Reed-Solomon, he states that a generator polynomial must be minimal, and that the coefficients of X^(n-k) and X^0 mnust be 1. Other texts define the generator polynomial as the Galois product of (X - alpha^m) .... (X - alpha^(m+2t-1)) where t is the number of errors and m is the starting location for the generator roots. If you take m = 2, the Galois product (X - alpha^2) ... (X - alpha^7), you will not get a 1 as the coefficient of the X^0 term. So the statement by Sklar and the definition in some books seem to differ, and information is needed on how to resolve this. Is there anything different about the encoder when the generator polynomial is defined to start at a location > 1?

Reply by ●October 9, 20082008-10-09

> If you take m = 2, ... > you will not get a 1 as the coefficient of the X^0 term. > ... information is needed on how to resolve this.The number is chosen to simplify the decoder. "1" and "0" are magic for Forney. With "1" the factor "x" following the division disappears. With "0" the factor changes to "1/x" and compensates a shift. Other magic numbers like in the NASA CCSDS were chosen for very special decoder-architectures ( Berlekamps bit-serial, dual-basis ... ). > the definition in some books seem to differ, Good books that cover RS very well are: Blahut "Theory and practice of error control codes" Addison Wesley 1983 Clark Cain "Error-Correction Coding for digital Communications" Plenum 1981 Wicker "Error Control Systems" Prentice Hall 1995 Perhaps available at the local library. MfG JRD

Reply by ●October 9, 20082008-10-09

On Oct 9, 11:55�am, "lindasel" <lselt...@alumni.caltech.edu> wrote:> Is there also a paper on the encoder?There are lots of papers on encoders, though I have not been fortunate enough to write any of them.> I have the following question about the encoder. > In Bernard Sklar's paper on what communications engineers should know > about Reed-Solomon, he states that a generator polynomial must be minimal, > and that the coefficients of X^(n-k) and X^0 mnust be 1.I don't know which paper this is, but the requirement that "the coefficients of X^(n-k) and X^0 mnust be 1" is not applicable to Reed-Solomon codes.> Other texts define the generator polynomial as the Galois product of > (X - alpha^m) .... (X - alpha^(m+2t-1)) > where t is the number of errors and m is the starting location for the > generator roots.This is correct.> If you take m = 2, the Galois product > (X - alpha^2) ... (X - alpha^7), > you will not get a 1 as the coefficient of the X^0 term. > So the statement by Sklar and the definition in some books seem to differ, > and information is needed on how to resolve this.You will not get 1 as the coefficient of the X^0 term even when m = 1. If Sklar did make the statement that you attribute to him, then that statement is wrong. Eq. (22) of his paper titled Reed-Solomon Codes (www.informit.com/content/images/art_sklar7_reed-solomon/elementLinks/ art_sklar7_reed-solomon.pdf) shows a generator polynomial with the coefficient of X^0 not being 1.> Is there anything different about the encoder when the generator > polynomial is defined to start at a location > 1?The answer can be implementation-dependent but there is not anything significantly different. In the *decoder*, as Rafael Deliano points out in another message in this thread, choosing m = 0 or 1 does have some advantages. --Dilip Sarwate

Reply by ●October 13, 20082008-10-13

>I won't vouch for the reliability or accuracy of the information >but you might want to look at the paper High-Speed Architectures for >Reed-Solomon Decoders, IEEE Transactions on VLSI Systems, vol. 9, >no. 5, Oct 2001 which is available (in Postscript format) at >http://www.ifp.uiuc.edu/~sarwate/decoder.ps >--Dilip SarwateTo Professor Sarwate: Thank you for posting your paper on your website. I would appreciate it if you could answer a question about the iBM algorithm. The main iteration loop is for r=0 step 1 until (2t-1) do Question 1: Is the "until" inclusive or noninclusive. Question 2: If the subscript of s can be r-0, r-1, ... r-t, then there would be cases in which the subscript of s is a negative number. Also, wouldn't there be a different number of terms for each iteration of r? Can you provide another formulation of this? Also do you have a worked example, particularly of the iBM algorithm? I would like to check my code at each step. If someone else has this working, your information would be appreciated.

Reply by ●October 15, 20082008-10-15

>of his paper titled Reed-Solomon Codes >(www.informit.com/content/images/art_sklar7_reed-solomon/elementLinks/ >art_sklar7_reed-solomon.pdf) >shows a generator polynomial with the coefficient of X^0 >not being 1.This link gave me an error message. The site seems to have restricted access.

Reply by ●October 15, 20082008-10-15

> iBM I can´t help you with the inversionless BM only with the old fashioned one. > I would like to check my code at each step. For that you will have to code the subroutines to do the galois-field and polynomial math. Most textbooks start out with a simple 4 bit field ( g = x^4 + x + 1 ) that will be sufficient for BCH(15,7) BCH(15,5) RS(15,9) RS(15,11). Assuming that works one can scale up to 8 bit. My description is in german, but probably clear enough if you have any book to help you. The calculation for the "digits": http://www.embeddedforth.de/temp/gal.pdf Ignore pictures 2,3,6,7: they are for implementation in hardware. The Lin Costello book has a good description of that. Ignore Zech: not necessary. View the result as a closed "galois-field-coprocessor" ( lower half of page ): http://www.embeddedforth.de/temp/fr.pdf Note that tables for squarer, cuber, inversion are handy. The upper half of the pages shows the BCH-decoder: somewhat simpler because binary and therefore no need for Forney. The next layer is polynomials ( "strings of digits" ). View the result as a closed "polynomial-processor". Skip division you do not need it for BM. http://www.embeddedforth.de/temp/pr.pdf Ignore the half page Euclid. Ignore the first 2 pages on Euclid: http://www.embeddedforth.de/temp/bm.pdf But note at picture 14,15 that two notations for S(z) are common: either you have a dummy "1" as first digit or not. Many flowcharts for BM on page 3,4. They are all slightly different improvements on Massey, they all worked for me. I prefer Blahut ( he has the best description on BM in his book ). Table 4 has detailed runs, Table 5 only I/O-data. Stuff like "L(z)-d*z*B(z)" is in the "polynomial-processor": shift ( picture 9 ) multiplication of constant ( picture 3 ) addition ( picture 2 ) "L(z)/d" is: division by constant ( picture 4 ) Most foulups in coding are in calculating discrepancy d due to the different notations for S(z) The simple BS will work for BCH but needs for RS an external Omega-calculator ( page 5 ): http://www.embeddedforth.de/temp/rs.pdf Its a multiplication of polynomials, result chopped to the right length. My code probably won´t help you as it is in FORTH. You can buy the first edition of Morelos-Zaragoza "The Art of Error Correcting Coding" at ebay from the UK very cheaply at the moment: http://cgi.ebay.de/The-Art-of-Error-Correcting-Coding-Book_W0QQitemZ370094882240QQcmdZViewItem?hash=item370094882240 For this book he has C code on his homepage. The book ( and that seems to go for the 2. edition too ) only has 220 pages and tries to cover a lot of subjects. Not much explaining there, but worked examples and at that price its worth having. MfG JRD

Reply by ●October 15, 20082008-10-15

On Oct 15, 2:40�am, "lindasel" <lselt...@alumni.caltech.edu> wrote:> >(www.informit.com/content/images/art_sklar7_reed-solomon/elementLinks/ > >art_sklar7_reed-solomon.pdf)> > This link gave me an error message. �The site seems to have restricted > access.The URL is very long, and Google Groups broke it into two lines. The URL that I posted (as opposed to what Google shows) continues past "...Links/" with the file name which ends in "solomon.pdf"