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i don't understand R & D White Paper WHP 031(page 26)

http://downloads.bbc.co.uk/rd/pubs/whp/whp-pdf-files/WHP031.pdf

why constant factor r=9. My calcuation is factor r=7

two polynomial are belows 7*(x^2)+ 7*(x^1)+9 , 3*(x^1)+14

Would you explain how 9 is extracted ?

I try to resolve using GF multiplication table and extened Euclid algorithm.

rcgldr
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user1395438
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2 Answers2

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The first polynomial, λΛ(x) = 7x^2+7x+9 is the error locator polynomial. The second polynomial, λΩ(x) = 3x+14, is used for calculating error values. Although not required, a common convention for Λ(x) is that the least significant term of Λ(x) is defined to be 1 (as opposed to the most significant term). In the example, the least significant term of λΛ(x) = 7x^2+7x+9 is the 9, so λ=9, and both polynomials are divided by λ=9. This results in (7x^2+7x+9)/9 = 14x^2+14x+1, and (3x+14)/9 = 6x+15.

A similar thing is shown in the wiki article, which uses GF(929), a non-binary field, in it's examples. In the wiki's extended Euclid example, both polynomials are divided by 544.

https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction#Euclidean_decoder

Comments about the white paper.

The Forney equation 21 on page 21 has an X^(1-b) term, which the paper notes can be ignored in the case b=1, but in the worked examples, b=0, so that X term is needed for the calculations as shown in the paper.

Some Reed Solomon implementations use a non-binary field, such as GF(929) as shown in the wiki article examples, and which is used for PDF417 barcodes. In this case, it's important to pay attention to addition versus subtraction, the signs of terms in equations, and the derivative of Λ(x) follows normal convention, the derivative of aX^b = abX^(b-1) with all math modulo 929. The wiki article examples show this.

rcgldr
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http://www.ujamjar.com/demo/ocaml/2014/06/18/reed-solomon-demo.html when m=6, k=2, b=1

more complicated polynomial extended euclid algorithm is done.

i just understand example in the white paper

but i test more complext example on the link.

i don't know why S4, S5

Would you explain how S4, S5 polynomial is extracted?

user1395438
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  • Sorry for the late response. Since you didn't comment on my answer, I didn't get any notification that you had added another answer. As for the demo parameters, I understand m=6 and b=1, but not k=2. Did you mean t=2? If so, there is no S4 or S5. If you set t=3, and create a 3 error case, then 6 syndromes, S0, S1, S2, S3, S4, S5 are used. – rcgldr Sep 20 '19 at 13:13