Talk:Reed–Solomon error correction

Shortened DVD

I am a new to Reed-solomon. Until recent,I am required to implement Reed-Solomon code on FPGAs. This will be used in our DVD project. However, I have some questions puzzling me greatly. In DVD error correction coding ,should RS(208,192,17) and RS(182,172,11) be shortened or puntured Reed-Solomon codes. I have no way to differentiate the two. Meanwhile, I have great eagerness to comminicate and discuss with others about ECC especially Reed-Solomon codes. My email is skycanny@hotmail.com

Regards!

Cross-interleaved coding

Is CIRC also used on DVDs.

Duality of the two views - inverse transform equation

Regarding the inverse transform equation for p(x), the equation is correct for non-binary fields $\textstyle v_{i}={\frac {1}{n}}q(\alpha ^{n-i})$, but for binary fields, the equation should be $\textstyle v_{i}=q(\alpha ^{n-i})$,

The issue depends if the field is a a non-binary field, such a a field modulo (929) as used in the examples versus a binary based field GF(2^n) where addition is effectively xor. As a simple way to explain this, for a non-binary field: $\sum _{i=1}^{n}1=n$ but for a binary field where n is an odd number $\sum _{i=1}^{n}1=1$ .

"Error correction" in title

Is there a good reason to include "error correction" in the title here? The current Applications section focuses on error correction, and those are probably the most notable applications, but there are some applications outside of correction, like key recovery, constructing MDS matrices, and some zero-knowledge argument protocols.

The article's scope already seems broader than error correction (e.g. it can correct up to t erasures); should we move it to "Reed-Solomon code" to reflect this broader scope? That would seem more WP:CONCISE and WP:CONSISTENT with Reed–Muller code, Algebraic geometry code, Expander code, etc. (which are also error-correcting codes). — xDanielx ^T/_C\^R 05:03, 29 July 2024 (UTC)

Duality of the two views

This section was removed, but I have found something similar. I'm leaving Ylloh's original comment:

I'm not sure what you are proposing here. The two views result in the same set of codewords, which is what is meant by "equivalence". It does not mean that the encoding functions are identical. Regarding the historical details, if you have more correct information on them, feel free to clarify. ylloh (talk) 19:59, 30 March 2016 (UTC)

Original view RS codewords have maximum length $n=q$, while BCH view codewords have maximum length $n=q-1$, and in this case, the codewords can not be the same. If original view uses sequential powers of the primitive element: $\{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{q-2}\}$ as evaluation points, then the resulting codeword of a set of values can be viewed as a set of coefficients to a polynomial $p(x)$, and the roots of $p(x)$ will be $\{\alpha ,\alpha ^{2},\dots ,\alpha ^{n-k}\}$. [Dual view of Reed Solomon codes]. If using systematic encoding for both original view and BCH view, original view encoding produces the same codewords as BCH view using generator polynomial $g(x)=(x-\alpha )(x-\alpha ^{2})\dots (x-\alpha ^{n-k})$. If using non-systematic encoding the roots will be the same, but not the codewords. However while original view implementations are normally least significant term first, BCH view implementations are normally most significant term first. In this case, to produce a codeword identical to original view systematic encoding, the roots are inverted, and the generator polynomial is $g(x)=(x-\alpha ^{k})(x-\alpha ^{k+1})\dots (x-\alpha ^{q-2})$. Rcgldr (talk) 16:45, 7 February 2025 (UTC)

History

Systematic encoding dates back to 1949 with Golay codes and 1950 with Hamming codes. Reed Solomon's 1960 paper describes a non-systematic encoding, and at that time, decoding required interpolation of all $nCk$ combinations of $n$ codeword symbols taken $k$ at a time and assuming the most popular result was the correct one. At about the same time, a practical decoder was invented for BCH codes, the Peterson Gorenstein Zierler decoder. Original view encoding was abandoned, and a switch to BCH view compatible encoding occurred. The textbook Error Correction Codes by Peterson and Weldon (1961), the first book about error correction codes, mentions RS codes as a sub-class of BCH codes, with systematic encoding, and there is no mention of the original view encoding. That textbook also covers Golay and Hamming codes. Original view systematic encoding is currently used for list decoders and some erasure codes. Note that with the constraints mentioned in Talk:Reed–Solomon_error_correction#Duality_of_the_two_views, original view encoding can be identical to BCH view encoding. Rcgldr (talk) 17:06, 7 February 2025 (UTC)