Generalized minimum-distance decoding: Difference between revisions

Revision as of 19:53, 15 November 2017

Revision as of 16:23, 9 November 2020

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: <math>\mathbb{E} \left [2X{_i^e + X{_i^?}} \right ] \leqslant {2e_i \over d}\qquad\qquad (2)</math>

Clearly, by definition

:<math>e' = \sum_i X_i^e \quad \text{and} \quad s' = \sum_i X_i^?.</math>

Further, by the [[linear]]ity of expectation, we get

:<math>\mathbb{E}[2e' + s'] \leqslant \frac{2}{d}\sum_ie_i < D.</math>

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==See also==

#* [[Concatenated code]]s

#* [[Reed Solomon|Reed Solomon error correction]]

#* [[Berlekamp–Welch algorithm|Welch Berlekamp algorithm]]

==References==

#* [http://www.cse.buffalo.edu/~atri/courses/coding-theory/lectures University at Buffalo Lecture Notes on Coding Theory – Atri Rudra]

#* [http://people.csail.mit.edu/madhu/FT01 MIT Lecture Notes on Essential Coding Theory – Madhu Sudan]

#* [http://www.cs.washington.edu/education/courses/cse533/06au University of Washington – Venkatesan Guruswami]

#* G. David Forney. Generalized Minimum Distance decoding. ''IEEE Transactions on Information Theory'', 12:125–131, 1966

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