Dynamic errors of numerical methods of ODE discretization: Difference between revisions

Latest revision as of 01:41, 19 November 2013

The Elias-Bassalygo bound is a mathematical limit used in coding theory for error correction during data transmission or communications. The properties of the Elias-Bassalygo bound are defined, below, using mathematical expressions.

Definition

Let $C$ be a $q$ -ary code of length $n$ , i.e. a subset of $[q]^{n}$ . (Each $q$ -ary block code of length $n$ is a subset of the strings of $𝒜_{q}^{n},$ where the alphabet set $𝒜_{q}$ has $q$ elements). Let $R$ be the rate of $C$ and $δ$ (delta) be the relative distance.

Let $B_{q} (y, ρ n) = {x \in [q]^{n} | Δ (x, y) \leq ρ n}$ be the Hamming ball of radius $ρ n$ centered at $y$ . Let $V o l_{q} (y, ρ n) = | B_{q} (y, ρ n) |$ be the volume of the Hamming ball of radius $ρ n$ . It is obvious that the volume of a Hamming Ball is translation-invariant, i.e. irrelevant with position of $y$ . In particular, $| B_{q} (y, ρ n) | = | B_{q} (0, ρ n) |$ .

With large enough $n$ , the rate $R$ and the relative distance $δ$ satisfies the Elias-Bassalygo bound: $R \leq 1 - H_{q} (J_{q} (δ)) + o (1),$

where

H_{q} (x) \equiv_{def} - x \cdot \log_{q} \frac{x}{q - 1} - (1 - x) \cdot \log_{q} (1 - x)

is the q-ary entropy function and

J_{q} (δ) \equiv_{def} (1 - \frac{1}{q}) (1 - \sqrt{1 - \frac{q δ}{q - 1}})

is a function related with Johnson bound.

Proof

To prove the Elias–Bassalygo bound, start with the following Lemma:

Lemma 1: Given a q-ary code, $C \subseteq [q]^{n}$ and $0 \leq e \leq n$ , there exists a Hamming ball of radius $e$ with at least $\frac{| C | V o l_{q} (0, e)}{q^{n}}$ codewords in it.

Proof of Lemma 1: To prove Lemma 1, use the probability method. Randomly pick a received word $y \in [q]^{n}$ . The expected size of overlapped region between $C$ and the Hamming ball centered at $y$ with radius $e$ , $| B_{q} (y, e) \cap C |$ is $V o l_{q} (y, e) \frac{| C |}{q^{n}}$ since $y$ is (uniform) randomly selected. Since this is the expected value of the size, there must exist at least one $y$ such that $| B_{q} (y, e) \cap C | \geq V o l_{q} (y, e) \frac{| C |}{q^{n}} = \frac{| C | V o l_{q} (0, e)}{q^{n}}$ , otherwise the expectation must be smaller than this value.

Now we prove the Elias–Bassalygo bound.

Define $e = n J_{q} (δ) - 1$ .

By Lemma 1, there exists a Hamming ball with $B$ codewords such that $B \geq \frac{| C | V o l (0, e)}{q^{n}}$

By the Johnson bound, we have $B \leq q d n$ . Thus,

∣ C ∣ \leq q n d \cdot \frac{q^{n}}{V o l_{q} (0, e)} \leq q^{n (1 - H_{q} (J_{q} (δ)) + o (1))}

The second inequality follows from lower bound on the volume of a Hamming ball: $V o l_{q} (0, ⌊ \frac{d - 1}{2} ⌋) \leq q^{H_{q} (\frac{δ}{2}) n - o (n)}$ .

Putting in $d = 2 e + 1$ and $δ = \frac{d}{n}$ gives the second inequality.

Therefore we have

R = \frac{\log_{q} | C |}{n} \leq 1 - H_{q} (J_{q} (δ)) + o (1)

References

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+The '''Elias-Bassalygo bound''' is a mathematical limit used in [[coding theory]] for [[error correction]] during [[data transmission]] or communications. The properties of the Elias-Bassalygo bound are defined, below, using mathematical expressions.<!--See "Hamming bound" for similar math-->
+== Definition ==
+Let <math>C</math> be a <math>q</math>-ary code of length <math>n</math>, i.e. a subset of <math>[q]^n</math>. (Each <math>q</math>-ary block code of length <math>n</math> is a subset of the strings of <math>\mathcal{A}_q^n\text{,}</math> where the alphabet set <math>\mathcal{A}_q</math> has <math>q</math> elements). Let <math>R</math> be the ''rate'' of <math>C</math> and <math>\delta</math> (delta) be the ''relative distance''.
+Let <math>B_q(\boldsymbol{y}, \rho n) =\{ \boldsymbol{x} \in [q]^n | \Delta(\boldsymbol{x}, \boldsymbol{y}) \le \rho n \}</math> be the ''[[Hamming ball]]'' of radius <math> \rho n </math> centered at <math>\boldsymbol{y}</math>.  Let <math> Vol_q(\boldsymbol{y}, \rho n) = |B_q(\boldsymbol{y}, \rho n)| </math> be the ''volume'' of the Hamming ball of radius <math> \rho n </math>.  It is obvious that the volume of a Hamming Ball is translation-invariant, i.e. irrelevant with position of <math>\boldsymbol{y}</math>.  In particular, <math> |B_q(\boldsymbol{y}, \rho n)| =|B_q(\boldsymbol{0}, \rho n)| </math>&nbsp;.
+With large enough <math>n</math>, the ''rate'' <math>R</math> and  the ''relative distance'' <math>\delta</math> satisfies the '''''Elias-Bassalygo bound:''''' <math>R \le 1 - H_q( J_q(\delta))+o(1)\text{,} </math>
+where
+: <math> H_q(x)\equiv_ \text{def} -x\cdot\log_q{x \over {q-1}}-(1-x)\cdot\log_q{(1-x)} </math>
+is the ''q''-ary entropy function
+and
+: <math>J_q(\delta) \equiv_ \text{def} \left(1-{1\over q}\right)\left(1-\sqrt{1-{q \delta \over{q-1}}}\right) </math> is a function related with [[Johnson bound]].
+== Proof ==
+To prove the Elias–Bassalygo bound, start with the following Lemma:
+'''''Lemma 1:''''' Given a ''q''-ary code, <math>C\subseteq  [q]^n </math> and <math> 0\le  e\le  n</math>, there exists a Hamming ball of radius <math>e</math> with at least <math>{|C|Vol_q(0,e)} \over {q^n}</math> codewords in it.
+Proof of Lemma 1: To prove ''Lemma 1'', use the probability method. Randomly pick a received word <math>y \in [q]^n</math>. The expected size of overlapped region between <math> C</math> and the Hamming ball centered at <math>y</math> with radius <math>e</math>, <math>|B_q(y,e) \cap C|</math> is <math>Vol_q(y,e) {{|C|} \over {q^n}}</math> since <math>y</math> is (uniform) randomly selected. Since this is the expected value of the size, there must exist at least one <math>y</math> such that <math>|B_q(y,e) \cap C| \ge Vol_q(y,e) {{|C|} \over {q^n}} = {{|C|Vol_q(0,e)} \over {q^n}}</math>, otherwise the expectation must be smaller than this value.
+Now we prove the Elias–Bassalygo bound.
+Define <math>e = n J_q(\delta)-1 </math>.
+By Lemma 1, there exists a Hamming ball with <math>B</math> codewords such that
+<math>B\ge { {|C|Vol(0,e)} \over {q^n}} </math>
+By the [[Johnson bound]], we have <math>B\le qdn</math>. Thus,
+: <math>\mid C \mid \le qnd \cdot {{q^n} \over {Vol_q(0,e)}} \le q^{n(1-H_q(J_q(\delta))+o(1))}</math>
+The second inequality follows from lower bound on the volume of a Hamming ball:
+<math> Vol_q(0, \lfloor {{d-1} \over 2} \rfloor) \le q^{H_q({\delta \over 2})n-o(n)} </math>.
+Putting in <math>d=2e+1</math> and <math> \delta = {d \over n}</math> gives the second inequality.
+Therefore we have
+: <math>R={\log_q{|C|} \over n} \le 1-H_q(J_q(\delta))+o(1) </math>
+== See also ==
+* [[Singleton bound]]
+* [[Hamming bound]]
+* [[Plotkin bound]]
+* [[Gilbert–Varshamov bound]]
+* [[Johnson bound]]
+==References==
+{{Reflist}}
+{{Citation
+ | title = New upper boundes for error-correcting codes
+ | year = 1965
+ | author = Bassalygo, L. A.
+ | journal = Problems of Information Transmission
+ | pages = 32–35
+ | volume = 1
+ | issue = 1
+}}
+{{Citation
+ | title = Lower bounds to error probability for coding on discrete memoryless channels. Part I.
+ | year = 1967
+ | journal = Information and Control
+ | pages = 65–103
+ | volume = 10
+ | last1 = Claude E. Shannon     | first1 =  Robert G. Gallager
+ | last2 =  Berlekamp
+ | first2 =  Elwyn R.
+}}
+{{Citation
+ | title = Lower bounds to error probability for coding on discrete memoryless channels. Part II.
+ | year = 1967
+ | journal = Information and Control
+ | pages = 522–552
+ | volume = 10
+ | last1 = Claude E. Shannon     | first1 =  Robert G. Gallager
+ | last2 =  Berlekamp
+ | first2 =  Elwyn R.
+}}
+[[Category:Coding theory]]
+[[Category:Articles containing proofs]]

Dynamic errors of numerical methods of ODE discretization: Difference between revisions

Latest revision as of 01:41, 19 November 2013

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