Block code - Revision history

Kvng: unpiped links using script

2025-03-28T15:25:56Z

unpiped links using script

← Previous revision		Revision as of 15:25, 28 March 2025
Line 3:		Line 3:

	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.
	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, [[mathematician]]s, and [[computer ~~science\|computer scientists~~]] to study the limitations of ''all'' block codes in a unified way.		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, [[mathematician]]s, and [[computer scientist]]s to study the limitations of ''all'' block codes in a unified way.
	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.

David Eppstein: /* Error detection and correction properties */ Hamming ball

2025-02-28T21:32:23Z

Error detection and correction properties: Hamming ball

← Previous revision		Revision as of 21:32, 28 February 2025
Line 70:		Line 70:
	== Error detection and correction properties ==		== Error detection and correction properties ==

	A codeword <math>c \in \Sigma^n</math>could be considered as a point in the <math>n</math>-dimension space <math>\Sigma^n</math> and the code <math>\mathcal{C}</math> is the subset of <math>\Sigma^n</math>. A code <math>\mathcal{C}</math> has distance <math>d</math> means that <math>\forall c\in \mathcal{C}</math>, there is no other codeword in the ''Hamming ball'' centered at <math>c</math> with radius <math>d-1</math>, which is defined as the collection of <math>n</math>-dimension words whose ''[[Hamming distance]]'' to <math>c</math> is no more than <math>d-1</math>. Similarly, <math> \mathcal{C}</math> with (minimum) distance <math>d</math> has the following properties:		A codeword <math>c \in \Sigma^n</math>could be considered as a point in the <math>n</math>-dimension space <math>\Sigma^n</math> and the code <math>\mathcal{C}</math> is the subset of <math>\Sigma^n</math>. A code <math>\mathcal{C}</math> has distance <math>d</math> means that <math>\forall c\in \mathcal{C}</math>, there is no other codeword in the [[Hamming ball]] centered at <math>c</math> with radius <math>d-1</math>, which is defined as the collection of <math>n</math>-dimension words whose ''[[Hamming distance]]'' to <math>c</math> is no more than <math>d-1</math>. Similarly, <math> \mathcal{C}</math> with (minimum) distance <math>d</math> has the following properties:
	* <math> \mathcal{C}</math> can detect <math>d-1</math> errors : Because a codeword <math>c</math> is the only codeword in the Hamming ball centered at itself with radius <math>d-1</math>, no error pattern of <math>d-1</math> or fewer errors could change one codeword to another. When the receiver detects that the received vector is not a codeword of <math> \mathcal{C}</math>, the errors are detected (but no guarantee to correct).		* <math> \mathcal{C}</math> can detect <math>d-1</math> errors : Because a codeword <math>c</math> is the only codeword in the Hamming ball centered at itself with radius <math>d-1</math>, no error pattern of <math>d-1</math> or fewer errors could change one codeword to another. When the receiver detects that the received vector is not a codeword of <math> \mathcal{C}</math>, the errors are detected (but no guarantee to correct).
	* <math> \mathcal{C}</math> can correct <math>\textstyle\left\lfloor {{d-1} \over 2}\right\rfloor</math> errors. Because a codeword <math>c</math> is the only codeword in the Hamming ball centered at itself with radius <math>d-1</math>, the two Hamming balls centered at two different codewords respectively with both radius <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math> do not overlap with each other. Therefore, if we consider the error correction as finding the codeword closest to the received word <math>y</math>, as long as the number of errors is no more than <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math>, there is only one codeword in the hamming ball centered at <math>y</math> with radius <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math>, therefore all errors could be corrected.		* <math> \mathcal{C}</math> can correct <math>\textstyle\left\lfloor {{d-1} \over 2}\right\rfloor</math> errors. Because a codeword <math>c</math> is the only codeword in the Hamming ball centered at itself with radius <math>d-1</math>, the two Hamming balls centered at two different codewords respectively with both radius <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math> do not overlap with each other. Therefore, if we consider the error correction as finding the codeword closest to the received word <math>y</math>, as long as the number of errors is no more than <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math>, there is only one codeword in the hamming ball centered at <math>y</math> with radius <math>\textstyle\left \lfloor {{d-1} \over 2}\right \rfloor</math>, therefore all errors could be corrected.

Kvng: rm dup. endmatter formatting.

2025-02-01T15:38:21Z

rm dup. endmatter formatting.

← Previous revision		Revision as of 15:38, 1 February 2025
Line 1:		Line 1:
	{{Short description\|Family of error-correcting codes that encode data in blocks}}		{{Short description\|Family of error-correcting codes that encode data in blocks}}

	{{More footnotes needed\|date=February 2025}}		{{More footnotes needed\|date=February 2025}}

	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.
	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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.		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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.
Line 171:		Line 171:

	{{reflist}}		{{reflist}}
			{{refbegin}}

	{{Refimprove\|date=September 2008}}

	* {{cite book \| author=J.H. van Lint \| authorlink=Jack van Lint \| title=Introduction to Coding Theory \| edition=2nd \| publisher=Springer-Verlag \| series=[[Graduate Texts in Mathematics\|GTM]] \| volume=86 \| year=1992 \| isbn=3-540-54894-7 \| page=[https://archive.org/details/introductiontoco0000lint/page/31 31] \| url=https://archive.org/details/introductiontoco0000lint/page/31 }}		* {{cite book \| author=J.H. van Lint \| authorlink=Jack van Lint \| title=Introduction to Coding Theory \| edition=2nd \| publisher=Springer-Verlag \| series=[[Graduate Texts in Mathematics\|GTM]] \| volume=86 \| year=1992 \| isbn=3-540-54894-7 \| page=[https://archive.org/details/introductiontoco0000lint/page/31 31] \| url=https://archive.org/details/introductiontoco0000lint/page/31 }}
	* {{cite book \| author=F.J. MacWilliams \| authorlink=Jessie MacWilliams \|author2=N.J.A. Sloane \|authorlink2=Neil Sloane \| title=The Theory of Error-Correcting Codes \| url=https://archive.org/details/theoryoferrorcor0000macw \| url-access=registration \| publisher=North-Holland \| year=1977 \| isbn=0-444-85193-3 \| page=[https://archive.org/details/theoryoferrorcor0000macw/page/35 35]}}		* {{cite book \| author=F.J. MacWilliams \| authorlink=Jessie MacWilliams \|author2=N.J.A. Sloane \|authorlink2=Neil Sloane \| title=The Theory of Error-Correcting Codes \| url=https://archive.org/details/theoryoferrorcor0000macw \| url-access=registration \| publisher=North-Holland \| year=1977 \| isbn=0-444-85193-3 \| page=[https://archive.org/details/theoryoferrorcor0000macw/page/35 35]}}
	* {{cite book \| author=W. Huffman \|author2=V.Pless \| authorlink2=Vera Pless \| title= Fundamentals of error-correcting codes \| url=https://archive.org/details/fundamentalsofer0000huff \| url-access=registration \| publisher=Cambridge University Press \| year=2003 \| isbn=978-0-521-78280-7}}		* {{cite book \| author=W. Huffman \|author2=V.Pless \| authorlink2=Vera Pless \| title= Fundamentals of error-correcting codes \| url=https://archive.org/details/fundamentalsofer0000huff \| url-access=registration \| publisher=Cambridge University Press \| year=2003 \| isbn=978-0-521-78280-7}}
	* {{cite book \| author=S. Lin \|author2=D. J. Jr. Costello \| title= Error Control Coding: Fundamentals and Applications \| publisher=Prentice-Hall \| year=1983 \| isbn=0-13-283796-X}}		* {{cite book \| author=S. Lin \|author2=D. J. Jr. Costello \| title= Error Control Coding: Fundamentals and Applications \| publisher=Prentice-Hall \| year=1983 \| isbn=0-13-283796-X}}
			{{refend}}

	== External links ==		== External links ==

Kvng: Added {{More footnotes needed}} tag

2025-02-01T15:37:22Z

Added {{More footnotes needed}} tag

← Previous revision		Revision as of 15:37, 1 February 2025
Line 1:		Line 1:
	{{Short description\|Family of error-correcting codes that encode data in blocks}}		{{Short description\|Family of error-correcting codes that encode data in blocks}}

			{{More footnotes needed\|date=February 2025}}
	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.
	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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.		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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.

Tea2min: Undid revision 1268097909 by 2402:E000:412:C337:0:0:0:1 (talk)

2025-01-08T18:23:34Z

Undid revision 1268097909 by 2402:E000:412:C337:0:0:0:1 (talk)

← Previous revision		Revision as of 18:23, 8 January 2025
Line 24:		Line 24:

	=== The alphabet Σ ===		=== The alphabet Σ ===
	The data stream to be encoded is modeled as a [[string (computer science)\|string]] over some '''alphabet''' <math>\Sigma</math>. The size <math>\|\Sigma\|</math> of the alphabet is often written as <math>q</math>. If <math>q=2</math>, then the block code is called a ''binary'' block code. In many applications it is useful to consider <math>q</math> to be a [[prime power]]		The data stream to be encoded is modeled as a [[string (computer science)\|string]] over some '''alphabet''' <math>\Sigma</math>. The size <math>\|\Sigma\|</math> of the alphabet is often written as <math>q</math>. If <math>q=2</math>, then the block code is called a ''binary'' block code. In many applications it is useful to consider <math>q</math> to be a [[prime power]], and to identify <math>\Sigma</math> with the [[finite field]] <math>\mathbb F_q</math>.

	, and to identify <math>\Sigma</math> with the [[finite field]] <math>\mathbb F_q</math>.

	=== The message length ''k'' ===		=== The message length ''k'' ===

2402:E000:412:C337:0:0:0:1 at 04:11, 8 January 2025

2025-01-08T04:11:39Z

← Previous revision		Revision as of 04:11, 8 January 2025
Line 24:		Line 24:

	=== The alphabet Σ ===		=== The alphabet Σ ===
	The data stream to be encoded is modeled as a [[string (computer science)\|string]] over some '''alphabet''' <math>\Sigma</math>. The size <math>\|\Sigma\|</math> of the alphabet is often written as <math>q</math>. If <math>q=2</math>, then the block code is called a ''binary'' block code. In many applications it is useful to consider <math>q</math> to be a [[prime power]], and to identify <math>\Sigma</math> with the [[finite field]] <math>\mathbb F_q</math>.		The data stream to be encoded is modeled as a [[string (computer science)\|string]] over some '''alphabet''' <math>\Sigma</math>. The size <math>\|\Sigma\|</math> of the alphabet is often written as <math>q</math>. If <math>q=2</math>, then the block code is called a ''binary'' block code. In many applications it is useful to consider <math>q</math> to be a [[prime power]]

			, and to identify <math>\Sigma</math> with the [[finite field]] <math>\mathbb F_q</math>.

	=== The message length ''k'' ===		=== The message length ''k'' ===

Kvng: request citation for unstable information

2024-12-13T14:40:33Z

request citation for unstable information

← Previous revision		Revision as of 14:40, 13 December 2024
Line 45:		Line 45:
	Then the minimum distance <math>d</math> of the code <math>C</math> is defined as		Then the minimum distance <math>d</math> of the code <math>C</math> is defined as
	:<math>d := \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)]</math>.		:<math>d := \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)]</math>.
	Since any code has to be [[injective]], any two codewords will disagree in at least one position, so the distance of any code is at least <math>1</math>. Besides, the '''distance''' equals the '''[[Hamming weight#Minimum weight\|minimum weight]]''' for linear block codes because:		Since any code has to be [[injective]], any two codewords will disagree in at least one position, so the distance of any code is at least <math>1</math>. Besides, the '''distance''' equals the '''[[Hamming weight#Minimum weight\|minimum weight]]''' for linear block codes because:{{cn\|date=December 2024}}
	:<math>\min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)] = \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[\mathbf{0},C(m_2)-C(m_1)] = \min_{m\in\Sigma^k\atop m\neq\mathbf{0}} w[C(m)] = w_\min</math>.		:<math>\min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)] = \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[\mathbf{0},C(m_2)-C(m_1)] = \min_{m\in\Sigma^k\atop m\neq\mathbf{0}} w[C(m)] = w_\min</math>.

37.110.210.98: Counterintuitive, possibly incorrect definition/proof of minimal distance being equal to minimal weight.

2024-12-07T20:19:45Z

Counterintuitive, possibly incorrect definition/proof of minimal distance being equal to minimal weight.

← Previous revision		Revision as of 20:19, 7 December 2024
Line 46:		Line 46:
	:<math>d := \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)]</math>.		:<math>d := \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)]</math>.
	Since any code has to be [[injective]], any two codewords will disagree in at least one position, so the distance of any code is at least <math>1</math>. Besides, the '''distance''' equals the '''[[Hamming weight#Minimum weight\|minimum weight]]''' for linear block codes because:		Since any code has to be [[injective]], any two codewords will disagree in at least one position, so the distance of any code is at least <math>1</math>. Besides, the '''distance''' equals the '''[[Hamming weight#Minimum weight\|minimum weight]]''' for linear block codes because:
	:<math>\min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)] = \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[\mathbf{0},C(~~m_1~~)+C(~~m_2~~)] = \min_{m\in\Sigma^k\atop m\neq\mathbf{0}} w[C(m)] = w_\min</math>.		:<math>\min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[C(m_1),C(m_2)] = \min_{m_1,m_2\in\Sigma^k\atop m_1\neq m_2} \Delta[\mathbf{0},C(m_2)-C(m_1)] = \min_{m\in\Sigma^k\atop m\neq\mathbf{0}} w[C(m)] = w_\min</math>.

	A larger distance allows for more error correction and detection.		A larger distance allows for more error correction and detection.

Kvng: WP:ADOPTYPO boolean -> Boolean

2024-11-14T00:28:58Z

WP:ADOPTYPO boolean -> Boolean

← Previous revision		Revision as of 00:28, 14 November 2024
Line 1:		Line 1:
	{{Short description\|Family of error-correcting codes that encode data in blocks}}		{{Short description\|Family of 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.		In [[coding theory]], '''block codes''' are a large and important family of [[Channel coding\|error-correcting codes]] that encode data in blocks.
	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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.		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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.
	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.

	Examples of block codes are [[Reed–Solomon code]]s, [[Hamming code]]s, [[Hadamard code]]s, [[Expander code]]s, [[Golay code (disambiguation)\|Golay code]]s, [[Reed–Muller code]]s and [[Polar code (coding theory)\|Polar 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.		Examples of block codes are [[Reed–Solomon code]]s, [[Hamming code]]s, [[Hadamard code]]s, [[Expander code]]s, [[Golay code (disambiguation)\|Golay code]]s, [[Reed–Muller code]]s and [[Polar code (coding theory)\|Polar 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.

	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}}

141.30.119.186: Add polar code to list of examples.

2024-06-05T09:03:57Z

Add polar code to list of examples.

← Previous revision		Revision as of 09:03, 5 June 2024
Line 4:		Line 4:
	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.

	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.		Examples of block codes are [[Reed–Solomon code]]s, [[Hamming code]]s, [[Hadamard code]]s, [[Expander code]]s, [[Golay code (disambiguation)\|Golay code]]s, [[Reed–Muller code]]s and [[Polar code (coding theory)\|Polar 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.

	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}}

← Previous revision		Revision as of 15:38, 1 February 2025
Line 1:		Line 1:
	{{Short description\|Family of error-correcting codes that encode data in blocks}}		{{Short description\|Family of error-correcting codes that encode data in blocks}}

	{{More footnotes needed\|date=February 2025}}		{{More footnotes needed\|date=February 2025}}

	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.
	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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.		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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.
Line 171:		Line 171:

	{{reflist}}		{{reflist}}
			{{refbegin}}

	{{Refimprove\|date=September 2008}}

	* {{cite book \| author=J.H. van Lint \| authorlink=Jack van Lint \| title=Introduction to Coding Theory \| edition=2nd \| publisher=Springer-Verlag \| series=[[Graduate Texts in Mathematics\|GTM]] \| volume=86 \| year=1992 \| isbn=3-540-54894-7 \| page=[https://archive.org/details/introductiontoco0000lint/page/31 31] \| url=https://archive.org/details/introductiontoco0000lint/page/31 }}		* {{cite book \| author=J.H. van Lint \| authorlink=Jack van Lint \| title=Introduction to Coding Theory \| edition=2nd \| publisher=Springer-Verlag \| series=[[Graduate Texts in Mathematics\|GTM]] \| volume=86 \| year=1992 \| isbn=3-540-54894-7 \| page=[https://archive.org/details/introductiontoco0000lint/page/31 31] \| url=https://archive.org/details/introductiontoco0000lint/page/31 }}
	* {{cite book \| author=F.J. MacWilliams \| authorlink=Jessie MacWilliams \|author2=N.J.A. Sloane \|authorlink2=Neil Sloane \| title=The Theory of Error-Correcting Codes \| url=https://archive.org/details/theoryoferrorcor0000macw \| url-access=registration \| publisher=North-Holland \| year=1977 \| isbn=0-444-85193-3 \| page=[https://archive.org/details/theoryoferrorcor0000macw/page/35 35]}}		* {{cite book \| author=F.J. MacWilliams \| authorlink=Jessie MacWilliams \|author2=N.J.A. Sloane \|authorlink2=Neil Sloane \| title=The Theory of Error-Correcting Codes \| url=https://archive.org/details/theoryoferrorcor0000macw \| url-access=registration \| publisher=North-Holland \| year=1977 \| isbn=0-444-85193-3 \| page=[https://archive.org/details/theoryoferrorcor0000macw/page/35 35]}}
	* {{cite book \| author=W. Huffman \|author2=V.Pless \| authorlink2=Vera Pless \| title= Fundamentals of error-correcting codes \| url=https://archive.org/details/fundamentalsofer0000huff \| url-access=registration \| publisher=Cambridge University Press \| year=2003 \| isbn=978-0-521-78280-7}}		* {{cite book \| author=W. Huffman \|author2=V.Pless \| authorlink2=Vera Pless \| title= Fundamentals of error-correcting codes \| url=https://archive.org/details/fundamentalsofer0000huff \| url-access=registration \| publisher=Cambridge University Press \| year=2003 \| isbn=978-0-521-78280-7}}
	* {{cite book \| author=S. Lin \|author2=D. J. Jr. Costello \| title= Error Control Coding: Fundamentals and Applications \| publisher=Prentice-Hall \| year=1983 \| isbn=0-13-283796-X}}		* {{cite book \| author=S. Lin \|author2=D. J. Jr. Costello \| title= Error Control Coding: Fundamentals and Applications \| publisher=Prentice-Hall \| year=1983 \| isbn=0-13-283796-X}}
			{{refend}}

	== External links ==		== External links ==

← Previous revision		Revision as of 15:37, 1 February 2025
Line 1:		Line 1:
	{{Short description\|Family of error-correcting codes that encode data in blocks}}		{{Short description\|Family of error-correcting codes that encode data in blocks}}

			{{More footnotes needed\|date=February 2025}}
	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.
	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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.		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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.

← Previous revision		Revision as of 00:28, 14 November 2024
Line 1:		Line 1:
	{{Short description\|Family of error-correcting codes that encode data in blocks}}		{{Short description\|Family of 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.		In [[coding theory]], '''block codes''' are a large and important family of [[Channel coding\|error-correcting codes]] that encode data in blocks.
	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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.		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, [[mathematician]]s, and [[computer science\|computer scientists]] to study the limitations of ''all'' block codes in a unified way.
	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.

	Examples of block codes are [[Reed–Solomon code]]s, [[Hamming code]]s, [[Hadamard code]]s, [[Expander code]]s, [[Golay code (disambiguation)\|Golay code]]s, [[Reed–Muller code]]s and [[Polar code (coding theory)\|Polar 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.		Examples of block codes are [[Reed–Solomon code]]s, [[Hamming code]]s, [[Hadamard code]]s, [[Expander code]]s, [[Golay code (disambiguation)\|Golay code]]s, [[Reed–Muller code]]s and [[Polar code (coding theory)\|Polar 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.

	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}}