Generalized minimum-distance decoding: Difference between revisions

Revision as of 15:41, 13 May 2023

Revision as of 01:36, 24 October 2023

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In [[coding theory]], '''generalized minimum-distance (GMD) decoding''' provides an efficient [[algorithm]] for decoding [[concatenated code]]s, which is based on using an [[error]]s-and-[[Erasure code|erasures]] decoder for the [[outer code]].

A [[Concatenated error correction code#Decoding concatenated codes|naive decoding algorithm]] for concatenated codes can not be an optimal way of decoding because it does not take into account the information that [[maximum likelihood decoding]] (MLD) gives. In other words, in the naive algorithm, inner received [[Code word (communication)|codeword]]s are treated the same regardless of the difference between their [[hamming distance]]s. Intuitively, the outer decoder should place higher confidence in symbols whose inner [[code|encodings]] are close to the received word. [[David Forney]] in 1966 devised a better algorithm called generalized minimum distance (GMD) decoding which makes use of those information better. This method is achieved by measuring confidence of each received codeword, and erasing symbols whose confidence is below a desired value. And GMD decoding algorithm was one of the first examples of [[soft-decision decoder]]s. We will present three versions of the GMD decoding algorithm. The first two will be [[randomized algorithm]]s while the last one will be a [[deterministic algorithm]].

==Setup==

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# Run errors and erasure algorithm for <math>C_\text{out}</math> on <math>\mathbf{y}'' = (y_1'', \ldots, y_N'')</math>.

'''Theorem 1.''' ''Let y be a received word such that there exists a [[Code word (communication)|codeword]]'' <math>\mathbf{c} = (c_1,\cdots, c_N) \in C_\text{out}\circ{C_\text{in}} \subseteq [q^n]^N</math> ''such that'' <math>\Delta(\mathbf{c}, \mathbf{y}) < \tfrac{Dd}{2}</math>. ''Then the deterministic GMD algorithm outputs'' <math>\mathbf{c}</math>.

Note that a [[Concatenated codes|naive decoding algorithm for concatenated codes]] can correct up to <math>\tfrac{Dd}{4}</math> errors.