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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 [[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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# [[Hamming distance]] : Given two [[Euclidean vector|vector]]s <math>u, v\in\sum^n</math> the Hamming distance between u and v, denoted by <math>\Delta(u, v)</math>, is defined to be the number of positions in which u and v differ.
# Minimum distance : Let <math>C\subseteq\sum^n</math> be a [[code]]. The minimum distance of code C is defined to be <math>d = \min{\Delta(c_1, c_2)}</math> where <math>c_1 \ne c_2 \in C</math>
# Code concatenation : Given <math>m = (m_1, \ldots, m_K) \in [Q]^K</math>, consider two codes which we call outer code and inner code <math>C_\text{out} = [Q]^K \rightarrow [Q]^N, C_\text{in} : [q]^k \rightarrow [q]^n</math>, and their distances are <math>D</math> and <math>d</math>. A concatenated code can be achieved by <math>C_\text{out} \circ C_\text{in} (m) = (C_\text{in} (C_\text{out} (m)_1), \ldots, C_\text{in} (C_\text{out} (m)_N ))</math> where <math>C_\text{out}(m) = ((C_\text{out} (m)_1, \ldots, (m)_N ))</math>. Finally we will take <math>C_\text{out}</math> to be [[Reed Solomon|RS code]], which has an errors and erasure decoder, and <math>K = O(\log{N})</math>, which in turn implies that MLD on the inner code will be poly(<math>N</math>) time.
# Maximum likelihood decoding(MLD) : MLD is a decoding method for error correcting codes, which outputs the codeword closest to the received word in Hamming distance. The MLD function denoted by <math>D_{MLD} : \sum^n \rightarrow C</math> is defined as follows. For every <math>y\in\sum_n</math>, <math>D_{MLD}(y) = \arg \min_{c \in C}\Delta(c, y)</math>.
# [[Probability density function]] : A [[probability distribution]] <math>\Pr[\bullet]</math> on a sample space <math>S</math> is a mapping from events of <math>S</math> to [[real number]]s such that <math>\Pr[A] \ge 0</math> for any event <math>A</math>, <math>\Pr[S] = 1</math>, and <math>\Pr[A \cup B] = \Pr[A] + \Pr[B]</math> for any two mutually exclusive events <math>A</math> and <math>B</math>
# [[Expected value]] : The expected value of a [[discrete random variable]] <math>X</math> is <math>\mathbb{E} = \sum_x\Pr[X = x]</math>.


==Randomized algorithm==
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Consider the received word <math>\mathbf{y} = (y_1,\ldots,y_N) \in [q^n]^N</math> which corrupted by [[noisy channel]]. The following is the algorithm description for the general case. In this algorithm, we can decode y by just declaring an erasure at every bad position and running the errors and erasure decoding algorithm for <math>C_\text{out}</math> on the resulting vector.


'''Randomized_Decoder'''
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<br />'''Given : '''<math>\mathbf{y} = (y_1,\dots,y_N) \in [q^n]^N</math>.
# For every <math>1 \le i \le N</math>, compute <math>y_i^\prime = MLD_{C_\text{in}}(y_i)</math>.
# Set <math>\omega_i = \min(\Delta(C_\text{in}(y_i^\prime), y_i), {d\over2})</math>.
# For every <math>1 \le i \le N</math>, repeat : With probability <math>2\omega_i \over d</math>, set <math>y_i^{\prime\prime} \leftarrow</math> ?, otherwise set <math>y_i^{\prime\prime} = y_i'</math>.
# Run errors and erasure algorithm for <math>C_\text{out}</math> on <math>\mathbf{y}^{\prime\prime} = (y_1^{\prime\prime}, \ldots, y_N^{\prime\prime})</math>.
 
'''Theorem 1.''' ''Let y be a received word such that there exists a [[codeword]]'' <math>\mathbf{c} = (c_1,\ldots, c_N) \in C_\text{out}\circ{C_\text{in}} \subseteq  [q^n]^N</math> ''such that'' <math>\Delta(\mathbf{c}, \mathbf{y}) < \frac{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>Dd \over 4</math> errors.
 
'''Lemma 1.''' ''Let the assumption in Theorem 1 hold. And if'' <math>\mathbf{y^{\prime\prime}}</math> ''has'' <math>e'</math> ''errors and'' <math>s'</math> ''erasures(when compared with'' <math>\mathbf{c}</math>'') after'' '''Step 1''', ''then'' <math>\mathbb{E}[2e' + s'] < D</math>.
 
If <math>2e' + s' < D</math>, then the algorithm in '''Step 2''' will output <math>\mathbf{c}</math>. The lemma above says that in expectation, this is indeed the case. Note that this is not enough to prove '''Theorem 1''', but can be crucial in developing future variations of the algorithm.
 
'''Proof of lemma 1.''' For every <math>1 \le i \le N</math>, define <math>e_i = \Delta(y_i, c_i)</math>. This implies that
 
<math>\sum_{i=1}^N e_i <  \frac{Dd}{2} \qquad\qquad (1)</math>
 
Next for every <math>1 \le i \le N</math>, we define two [[indicator variable]]s:
 
: <math>X{_i^?} = 1</math> iff <math>y_i^{\prime\prime} = ?,</math>
 
and
 
: <math>X{_i^e} = 1</math> iff <math>C_\text{in}(y_i^{\prime\prime}) \ne c_i</math>
 
and
 
: <math>y_i^{\prime\prime} \ne ?.</math>
 
We claim that we are done if we can show that for every <math>1 \le i \le N</math>:
 
: <math>\mathbb{E}[2X{_i^e + X{_i^?}}] \le {{2e_i} \over d}\qquad\qquad (2)</math>
 
Clearly, by definition <math>e' = \sum_{i}X{_i^e}</math> and <math>s' = \sum_{i}X{_i^?}</math>. Further, by the [[linear]]ity of expectation, we get <math>\mathbb{E}[2e' + s'] \le {2 \over d}\sum_ie_i < D</math>. We consider two cases to prove (2) : <math>i'th</math> block is correctly decoded('''Case 1'''),  <math>i'th</math> block is incorrectly decoded('''Case 2''')
 
'''Case 1:''' <math>(c_i = C_\text{in}(y_i'))</math>
 
Note that if <math>y_i^{\prime\prime} = ?</math> then <math>X_i^e = 0</math>, and <math>\Pr[y_i^{\prime\prime} = ?] = {2\omega_i \over d}</math> implies
<math>\mathbb{E}[X_i^?] = \Pr[X_i^? = 1] = {2\omega_i \over d}</math>, and <math>\mathbb{E}[X_i^e] = \Pr[X_i^e = 1] = 0</math>.
 
Further, by definition we have
 
: <math>\omega_i = \min(\Delta(C_\text{in}(y_i'), y_i), {d \over 2}) \le \Delta(C_\text{in}(y_i'), y_i) = \Delta(c_i, y_i) = e_i</math>
 
'''Case 2:''' <math>(c_i \ne C_\text{in}(y_i'))</math>
 
In this case,
 
: <math>\mathbb{E}[X_i^?] = {2\omega_i \over d}</math>and <math>\mathbb{E}[X_i^e] = \Pr[X_i^e = 1] = 1 - {2\omega_i \over d}.</math>
 
Since <math>c_i \ne C_\text{in}(y_i')</math>, <math>e_i + \omega_i \ge d</math>. This follows [http://www.cse.buffalo.edu/~atri/courses/coding-theory/lectures/lect28.pdf another case analysis] when <math>(\omega_i = \Delta(C_\text{in}(y_i'),  y_i) < {d \over 2})</math> or not.
 
Finally, this implies
 
: <math>\mathbb{E}[2X_i^e + X_i^?] = 2 - {2\omega_i \over d} \le {2e_i \over d}.</math>
 
In the following sections, we will finally show that the deterministic version of the algorithm above can do unique decoding of <math>C_\text{out} \circ C_\text{in}</math> up to half its design distance.
 
==Modified randomized algorithm==
Note that, in the previous version of the GMD algorithm in step "3", we do not really need to use "fresh" [[randomness]] for each <math>i</math>. Now we come up with another randomized version of the GMD algorithm that uses the ''same'' randomness for every <math>i</math>. This idea follows the algorithm below.
 
'''Modified_Randomized_Decoder'''
<br />'''Given : '''<math>\mathbf{y} = (y_1, \ldots,y_N) \in [q^n]^N</math>, pick <math>\theta \in [0, 1]</math> at random. Then every for every <math>1 \le i \le N</math>:
# Set <math>y_i^\prime = MLD_{C_\text{in}}(y_i)</math>.
# Compute <math>\omega_i = \min(\Delta(C_\text{in}(y_i^\prime), y_i), {d\over2})</math>.
# If <math>\theta</math> < <math>{2\omega_i \over d}</math>, set <math>y_i^{\prime\prime} \leftarrow</math> ?, otherwise set <math>y_i^{\prime\prime} = y_i'</math>.
# Run errors and erasure algorithm for <math>C_\text{out}</math> on <math>\mathbf{y}^{\prime\prime} = (y_1^{\prime\prime},\ldots, y_N^{\prime\prime})</math>.
 
For the proof of '''[[Lemma (mathematics)|Lemma 1]]''', we only use the randomness to show that
 
: <math>\Pr[y_i^{\prime\prime} = ?] = {2\omega_i \over d}.</math>
 
In this version of the GMD algorithm, we note that
 
: <math>\Pr[y_i^{\prime\prime} = ?] = \Pr[\theta \in [0, {2\omega_i \over d}]] = {2\omega_i \over d}.</math>
 
The second [[Equality (mathematics)|equality]] above follows from the choice of <math>\theta</math>. The proof of '''Lemma 1''' can be also used to show <math>\mathbb{E}[2e' + s']</math> < <math>D</math> for version2 of GMD.
In the next section, we will see how to get a deterministic version of the GMD algorithm by choosing θ from a polynomially sized set as opposed to the current infinite set <math>[0, 1]</math>.
 
==Deterministic algorithm==
Let <math>Q = \{0,1\} \cup \{{2\omega_1 \over d}, \ldots,{2\omega_N \over d}\}</math>. Since for each <math>i, \omega_i = \min(\Delta(\mathbf{y_i'}, \mathbf{y_i}), {d \over 2})</math>, we have
 
: <math>Q = \{0, 1\} \cup \{q_1, \ldots,q_m\}</math>
 
where <math>q_1 < q_2 < \cdots < q_m</math> for some <math>m \le \left \lfloor \frac{d}{2} \right \rfloor</math>. Note that for every <math>\theta \in [q_i, q_{i+1}]</math>, the step 1 of the second version of randomized algorithm outputs the same <math>\mathbf{y^{\prime\prime}}</math>. Thus, we need to consider all possible value of <math>\theta \in Q</math>. This gives the deterministic algorithm below.
 
'''Deterministic_Decoder'''
<br />'''    Given : '''<math>\mathbf{y} = (y_1,\ldots,y_N) \in [q^n]^N</math>, for every <math>\theta \in Q</math>, repeat the following.
# Compute <math>y_i^\prime = MLD_{C_\text{in}}(y_i)</math> for <math>1 \le i \le N</math>.
# Set <math>\omega_i = \min(\Delta(C_\text{in}(y_i^\prime), y_i), {d\over2})</math> for every <math>1 \le i \le N</math>.
# If <math>\theta</math> < <math>{2\omega_i \over d}</math>, set <math>y_i^{\prime\prime} \leftarrow</math> ?, otherwise set <math>y_i^{\prime\prime} = y_i'</math>.
# Run errors-and-erasures algorithm for <math>C_\text{out}</math> on <math>\mathbf{y^{\prime\prime}} = (y_1^{\prime\prime}, \ldots, y_N^{\prime\prime})</math>. Let <math>c_\theta</math> be the codeword in <math>C_\text{out} \circ C_\text{in}</math> corresponding to the output of the algorithm, if any.
# Among all the <math>c_\theta</math> output in 4, output the one closest to <math>\mathbf{y}</math>
 
Every loop of 1~4 can be run in [[polynomial time]], the algorithm above can also be computed in polynomial time.
Specifically, each call to an errors and erasures decoder of <math><dD/2</math> errors takes <math>O(d)</math> time. Finally, the runtime of the algorithm above is <math>O(NQn^{O(1)} + NT_\text{out})</math> where <math>T_\text{out}</math> is the running time of the outer errors and erasures decoder.
 
==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
 
{{DEFAULTSORT:Generalized minimum distance decoding}}
[[Category:Error detection and correction]]
[[Category:Coding theory]]
[[Category:Finite fields]]
[[Category:Information theory]]

Latest revision as of 23:51, 5 May 2014


What's comprised within this write-up can alter your daily life. If you're considering investing, you require exactly what is supplied in this article, in order to be productive. If profits are what you desire and you will have the push and determination to help keep you experiencing both bad and good periods, keep reading to determine the key to accomplishment.

If you invest, be sure that you have reasonable objectives. Many individuals realize that if you do not be involved in heavy risk investing, that has a substantial potential for failing, you simply will not have good results using the marketplace over night. Bear this in mind as you may develop your stock portfolio to make sure you don't get taken good thing about.

Don't allow greed or eagerness overcome your judgements in terms of purchasing the stock exchange. Getting reduced and offering great is a very common hint simply because it seems sensible to acquire a supply when there's a higher opportunity which it will surge in value, even if you have to hold out for quite a while.

A stock's cost is not the only real sign of how costly it really is. Since carry, ideals are contingent upon profits, a stock which costs a hundred $ $ $ $ may possibly be affordable when the earnings' outlook is confident. Furthermore, a supply that costs just one or two dollars could be rather pricey in the event the connected company's earning projections usually are not brilliant.

Opt for the best stocks and shares in a number of market sectors to produce a properly-healthy collection. The entire industry will develop, but there are several areas that do not see any surge in development. To enhance your portfolio as a whole, you have to have stocks and shares through the market sectors that are increasing, and also this includes having shares from different market sectors. Normal collection re-managing can lessen any losses within-carrying out sectors, while getting you into others which can be presently growing.

When it comes to making an investment, be sure you're informed. Learn the basics of accounting and stock market historical past. If you're not informed, you won't be able to generate profits and you'll seem like a deceive. You don't need to have a 4 year data processing degree or nearly anything fancy, but take the time to discover the important information.

Change your border of basic safety in line with the reputation, earnings, and scale of a selected organization. While companies like Google or Johnson And Johnson are hardy and often stay around, there are particular businesses that may possibly do very well for quite a while just before crashing. Take this into account when selecting shares.

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Hold your shares provided that you can, from no less than 5yrs to maybe eternity. Usually do not offer once the marketplaces have been hard to get a day time or possibly a season. Also will not market when your carry has more than doubled or tripled. As long as your factors behind keeping that carry continue to be excellent, then keep positioning it. Reinvest any revenue you do not need over the following five years. Offer as long as the carry goes so high how the industry is just maxed out and not gonna expand anymore.

Should you be encouraged to always steer clear of stocks with astronomically higher financial debt-to-equity proportions, always keep this principle at heart by using a grain of sea salt. Even though it is a solid general guideline, a notable different does are available for circumstances brought on by share repurchases. In these cases, the debt-to-home equity rate has run out of normal positioning on account of supply buyback and needs a chance to proper.

Viewing a company's stock value progress up and comprehending why, is significantly different than knowing beforehand that you believe the business is on the rise. Discovering firms that seem poised to produce a relocate takes significant amounts of analysis. Also, analyst studies are good to look at, too. Do your research, and select businesses that you feel happen to be in progress setting.

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