From 654e7e10827b0dea7e692dde09a6d66afc846dd6 Mon Sep 17 00:00:00 2001 From: Christopher Haster Date: Wed, 30 Oct 2024 12:49:35 -0500 Subject: [PATCH] README.md - Removed junk after error+erasure section --- README.md | 137 ------------------------------------------------------ 1 file changed, 137 deletions(-) diff --git a/README.md b/README.md index 7fb46ac..50c912f 100644 --- a/README.md +++ b/README.md @@ -1514,140 +1514,3 @@ And some caveats: src="https://latex.codecogs.com/svg.image?C%28x%29%20%3d%20C%27%28x%29%20%2d%20%5csum_%7bj%20%5cin%20E%20%5ccup%20F%7d%20Y_j%20x%5ej" >

- - - vvv TODO vvv - -the algorithm as normal - -, which tells - us everything we need to know to find the location of errors and - erasures: - - - We can then continue the algorithm as normal, finding errors/erasures - where $\Lambda(X_j^{-1}) = 0$, and repairing them with Forney's - algorithm. - - - - - - - vvv TODO vvv - - Which has all of the properties of t - - - - - - Use Berlekamp-Massey to find the error-locator polynomial - $\Lambda_E(x)$ from the Forney syndromes $S_{Ei}$. - - - - - - - need to sort of hide the effects of our erasures from the - - In order to find the error-locator polynomial $\Lambda_E(x)$, - - - - - - Renaming $\Lambda(x)$ to the error-and-erasure-locator polynomial, we - can - - First note we can split the error-locator polynomial $\Lambda(x)$ into - separate - - - - - - - - - - vvv TODO vvv - - - - With Reed-Solomon, each unknown-location error requires 2 bytes of ECC - to find and repair, but each known-location error, usually called - "erasures", require only 1 byte of ECC to correct. - - - - - - - All of the above math assumes we don't know the location of errors, - since this is usually the case for block devices, and requires $2e$ - bytes of `ecc_size` to find $e$ errors. - - But if we know the location of errors, via parity bits or other - side-channels, we can actually do a bit better and find $e$ errors - with only $e$ bytes of `ecc_size`. We usually call these "erasures". - - You can even mix and match errors and erasures as long as you have - enough `ecc_size`, with each error needing 2 bytes of `ecc_size`, and - each erasure needing 1 bytes of `ecc_size`. - - TODO - - Find the erasure-locator polynomial $\Lambda_F(x)$: - -

- \Lambda_F(x) = \prod_{j \in F} \left(1 - X_j x\right) -

- - Note: - -

- \Lambda(x) = \Lambda_E(x) \Lambda_F(x) -

- - Find the Forney syndromes $S_{Ei}$: - -

- S_E(x) = S(x) \Lambda_F(x) \bmod x^n -

- - Note: - -

- \begin{aligned} \Omega(x) &= S(x)\Lambda(x) \bmod x^n \\ &= S(x)\Lambda_E(x)\Lambda_F(x) \bmod x^n \\ &= S_F(x)\Lambda_E(x) \bmod x^n \end{aligned} -

- - Use Berlekamp-Massey to find the error-locator polynomial - $\Lambda_E(x)$ from the Forney syndromes $S_{Ei}$. - - - Combine: - -

- \Lambda(x) = \Lambda_E(x) \Lambda_F(x) -

- - And continue as normal, finding $X_j$ where $\Lambda(X_j^{-1})=0$ - and solving for $Y_j$ where - $Y_j = X_j \frac{\Omega(X_j^{-1})}{\Lambda'(X_j^{-1})}$.