You are about to undo an edit. Please check the comparison below to verify that this is what you want to do, then publish the changes below to finish undoing the edit. If you are undoing an edit that is not vandalism, explain the reason in the edit summary. Do not use the default message only. |
Latest revision | Your text | ||
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. |
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> |
: <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: |
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>. |
:<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. |
[[Reed–Solomon code]]s are non-trivial examples of codes that satisfy the singleton bound with equality. |
||
=== |
===[[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>. |
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> |
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>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. |
<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 /> |
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}}. |
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> |
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> |
: <math>R={\log_q{|C|} \over n} \le 1-H_q\left(J_q\left(\delta\right)\right)+o\left(1\right) </math> |
||
Copy and paste: – — ° ′ ″ ≈ ≠ ≤ ≥ ± − × ÷ ← → · § Cite your sources: <ref></ref>
{{}} {{{}}} | [] [[]] [[Category:]] #REDIRECT [[]] <s></s> <sup></sup> <sub></sub> <code></code> <pre></pre> <blockquote></blockquote> <ref></ref> <ref name="" /> {{Reflist}} <references /> <includeonly></includeonly> <noinclude></noinclude> {{DEFAULTSORT:}} <nowiki></nowiki> <!-- --> <span class="plainlinks"></span>
Symbols: ~ | ¡ ¿ † ‡ ↔ ↑ ↓ • ¶ # ∞ ‹› «» ¤ ₳ ฿ ₵ ¢ ₡ ₢ $ ₫ ₯ € ₠ ₣ ƒ ₴ ₭ ₤ ℳ ₥ ₦ № ₧ ₰ £ ៛ ₨ ₪ ৳ ₮ ₩ ¥ ♠ ♣ ♥ ♦ 𝄫 ♭ ♮ ♯ 𝄪 © ® ™
Latin: A a Á á À à  â Ä ä Ǎ ǎ Ă ă Ā ā à ã Å å Ą ą Æ æ Ǣ ǣ B b C c Ć ć Ċ ċ Ĉ ĉ Č č Ç ç D d Ď ď Đ đ Ḍ ḍ Ð ð E e É é È è Ė ė Ê ê Ë ë Ě ě Ĕ ĕ Ē ē Ẽ ẽ Ę ę Ẹ ẹ Ɛ ɛ Ǝ ǝ Ə ə F f G g Ġ ġ Ĝ ĝ Ğ ğ Ģ ģ H h Ĥ ĥ Ħ ħ Ḥ ḥ I i İ ı Í í Ì ì Î î Ï ï Ǐ ǐ Ĭ ĭ Ī ī Ĩ ĩ Į į Ị ị J j Ĵ ĵ K k Ķ ķ L l Ĺ ĺ Ŀ ŀ Ľ ľ Ļ ļ Ł ł Ḷ ḷ Ḹ ḹ M m Ṃ ṃ N n Ń ń Ň ň Ñ ñ Ņ ņ Ṇ ṇ Ŋ ŋ O o Ó ó Ò ò Ô ô Ö ö Ǒ ǒ Ŏ ŏ Ō ō Õ õ Ǫ ǫ Ọ ọ Ő ő Ø ø Œ œ Ɔ ɔ P p Q q R r Ŕ ŕ Ř ř Ŗ ŗ Ṛ ṛ Ṝ ṝ S s Ś ś Ŝ ŝ Š š Ş ş Ș ș Ṣ ṣ ß T t Ť ť Ţ ţ Ț ț Ṭ ṭ Þ þ U u Ú ú Ù ù Û û Ü ü Ǔ ǔ Ŭ ŭ Ū ū Ũ ũ Ů ů Ų ų Ụ ụ Ű ű Ǘ ǘ Ǜ ǜ Ǚ ǚ Ǖ ǖ V v W w Ŵ ŵ X x Y y Ý ý Ŷ ŷ Ÿ ÿ Ỹ ỹ Ȳ ȳ Z z Ź ź Ż ż Ž ž ß Ð ð Þ þ Ŋ ŋ Ə ə
Greek: Ά ά Έ έ Ή ή Ί ί Ό ό Ύ ύ Ώ ώ Α α Β β Γ γ Δ δ Ε ε Ζ ζ Η η Θ θ Ι ι Κ κ Λ λ Μ μ Ν ν Ξ ξ Ο ο Π π Ρ ρ Σ σ ς Τ τ Υ υ Φ φ Χ χ Ψ ψ Ω ω {{Polytonic|}}
Cyrillic: А а Б б В в Г г Ґ ґ Ѓ ѓ Д д Ђ ђ Е е Ё ё Є є Ж ж З з Ѕ ѕ И и І і Ї ї Й й Ј ј К к Ќ ќ Л л Љ љ М м Н н Њ њ О о П п Р р С с Т т Ћ ћ У у Ў ў Ф ф Х х Ц ц Ч ч Џ џ Ш ш Щ щ Ъ ъ Ы ы Ь ь Э э Ю ю Я я ́
IPA: t̪ d̪ ʈ ɖ ɟ ɡ ɢ ʡ ʔ ɸ β θ ð ʃ ʒ ɕ ʑ ʂ ʐ ç ʝ ɣ χ ʁ ħ ʕ ʜ ʢ ɦ ɱ ɳ ɲ ŋ ɴ ʋ ɹ ɻ ɰ ʙ ⱱ ʀ ɾ ɽ ɫ ɬ ɮ ɺ ɭ ʎ ʟ ɥ ʍ ɧ ʼ ɓ ɗ ʄ ɠ ʛ ʘ ǀ ǃ ǂ ǁ ɨ ʉ ɯ ɪ ʏ ʊ ø ɘ ɵ ɤ ə ɚ ɛ œ ɜ ɝ ɞ ʌ ɔ æ ɐ ɶ ɑ ɒ ʰ ʱ ʷ ʲ ˠ ˤ ⁿ ˡ ˈ ˌ ː ˑ ̪ {{IPA|}}
Wikidata entities used in this page
Pages transcluded onto the current version of this page (help):
This page is a member of 9 hidden categories (help):