Editing Block code

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@@ Line 97: / Line 97: @@
 To explore the relationship between <math>R(C)</math> and <math>\delta(C)</math>, a set of lower and upper bounds of block codes are known.
-=== Hamming bound ===
+=== [[Hamming bound]] ===
-{{main article|Hamming bound}}
 : <math> R \le 1- {1 \over n} \cdot \log_{q} \cdot \left[\sum_{i=0}^{\left\lfloor {{\delta \cdot n-1}\over 2}\right\rfloor}\binom{n}{i}(q-1)^i\right]</math>
-=== Singleton bound ===
+=== [[Singleton bound]] ===
-{{main article|Singleton bound}}
 The Singleton bound is that the sum of the rate and the relative distance of a block code cannot be much larger than 1:
 :<math> R + \delta \le  1+\frac{1}{n}</math>.
@@ Line 108: / Line 106: @@
 [[Reed–Solomon code]]s are non-trivial examples of codes that satisfy the singleton bound with equality.
-=== Plotkin bound ===
+===[[Plotkin bound]]===
-{{main article|Plotkin bound}}
 For <math>q=2</math>, <math>R+2\delta\le1</math>. In other words, <math>k + 2d \le n</math>.
@@ Line 119: / Line 116: @@
 For any {{mvar|q}}-ary code with distance <math>\delta</math>, <math>R \le 1- \left({q \over {q-1}}\right) \delta + o\left(1\right)</math>
-=== Gilbert–Varshamov bound ===
+===[[Gilbert-Varshamov bound|Gilbert–Varshamov bound]]===
-{{main article|Gilbert–Varshamov bound}}
 <math>R\ge1-H_q\left(\delta\right)-\epsilon</math>, where <math>0 \le \delta \le 1-{1\over q}, 0\le \epsilon \le 1- H_q\left(\delta\right)</math>,
 <math> H_q\left(x\right) ~\overset{\underset{\mathrm{def}}{}}{=}~ -x\cdot\log_q{x \over {q-1}}-\left(1-x\right)\cdot\log_q{\left(1-x\right)} </math> is the {{mvar|q}}-ary entropy function.
-=== Johnson bound ===
+=== [[Johnson bound]] ===
-{{main article|Johnson bound}}
 Define <math>J_q\left(\delta\right) ~\overset{\underset{\mathrm{def}}{}}{=}~ \left(1-{1\over q}\right)\left(1-\sqrt{1-{q \delta \over{q-1}}}\right) </math>. <br />
 Let <math>J_q\left(n, d, e\right)</math> be the maximum number of codewords in a Hamming ball of radius {{mvar|e}} for any code <math>C \subseteq \mathbb{F}_q^n</math> of distance {{mvar|d}}.
@@ Line 131: / Line 126: @@
 Then we have the ''Johnson Bound'' : <math>J_q\left(n,d,e\right)\le qnd</math>, if <math>{e \over n} \le {{q-1}\over q}\left( {1-\sqrt{1-{q \over{q-1}}\cdot{d \over n}}}\, \right)=J_q\left({d \over n}\right)</math>
-=== Elias–Bassalygo bound ===
+=== [[Elias Bassalygo bound|Elias–Bassalygo bound]] ===
-{{main article|Elias Bassalygo bound}}
 : <math>R={\log_q{|C|} \over n} \le 1-H_q\left(J_q\left(\delta\right)\right)+o\left(1\right) </math>

Editing Block code

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