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{{Short description|Family of error-correcting codes that encode data in blocks}} |
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In [[coding theory]], '''block codes''' are a large and important family of [[Channel coding|error-correcting codes]] that encode data in blocks. |
In [[coding theory]], '''block codes''' are a large and important family of [[Channel coding|error-correcting codes]] that encode data in blocks. |
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There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, [[ |
There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, [[mathematics|mathematicians]], and [[computer science|computer scientists]] to study the limitations of ''all'' block codes in a unified way. |
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Such limitations often take the form of ''bounds'' that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors. |
Such limitations often take the form of ''bounds'' that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors. |
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Examples of block codes are [[Reed–Solomon code]]s, [[Hamming code]]s, [[Hadamard code]]s, [[Expander code]]s, [[Golay code (disambiguation)|Golay code]]s, |
Examples of block codes are [[Reed–Solomon code]]s, [[Hamming code]]s, [[Hadamard code]]s, [[Expander code]]s, [[Golay code (disambiguation)|Golay code]]s, and [[Reed–Muller code]]s. These examples also belong to the class of [[linear code]]s, and hence they are called '''linear block codes'''. More particularly, these codes are known as algebraic block codes, or cyclic block codes, because they can be generated using boolean polynomials. |
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Algebraic block codes are typically [[Soft-decision decoder|hard-decoded]] using algebraic decoders.{{Technical statement|date=May 2015}} |
Algebraic block codes are typically [[Soft-decision decoder|hard-decoded]] using algebraic decoders.{{Technical statement|date=May 2015}} |
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== Lower and upper bounds of block codes == |
== Lower and upper bounds of block codes == |
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[[File:HammingLimit.png|thumb|720px|Hamming limit |
[[File:HammingLimit.png|thumb|720px|Hamming limit]] |
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[[File:Linear Binary Block Codes and their needed Check Symbols.png|thumb|720px| |
[[File:Linear Binary Block Codes and their needed Check Symbols.png|thumb|720px| |
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There are theoretical limits (such as the Hamming limit), but another question is which codes can actually constructed. |
There are theoretical limits (such as the Hamming limit), but another question is which codes can actually constructed. It is like [[Sphere packing|packing spheres in a box]] in many dimensions. This diagram shows the constructible codes, which are linear and binary. The ''x'' axis shows the number of protected symbols ''k'', the ''y'' axis the number of needed check symbols ''n–k''. Plotted are the limits for different Hamming distances from 1 (unprotected) to 34. |
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Marked with dots are perfect codes: |
Marked with dots are perfect codes: |
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{{bulleted list |
{{bulleted list |
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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 ́
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