Block code: Difference between revisions

Revision as of 11:52, 18 April 2023

Latest revision as of 15:25, 28 March 2025

(12 intermediate revisions by 8 users not shown)

Line 1:

{{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.

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 scientistsscientist]]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.

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 [[Reed–MullerPolar 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 booleanBoolean polynomials.

Algebraic block codes are typically [[Soft-decision decoder|hard-decoded]] using algebraic decoders.{{Technical statement|date=May 2015}}

Line 13 ⟶ 15:

== The block code and its parameters ==

{| class="wikitable"

|+

!

!

!

!

|-

|

|

|

|

|-

|

|

|

|

|-

|

|

|

|

|}

[[Error-correcting code]]s are used to [[reliability (computer networking)|reliably]] transmit [[digital data]] over unreliable [[communication channel]]s subject to [[channel noise]].

When a sender wants to transmit a possibly very long data stream using a block code, the sender breaks the stream up into pieces of some fixed size. Each such piece is called ''message'' and the procedure given by the block code encodes each message individually into a codeword, also called a ''block'' in the context of block codes. The sender then transmits all blocks to the receiver, who can in turn use some decoding mechanism to (hopefully) recover the original messages from the possibly corrupted received blocks.

Line 65 ⟶ 46:

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>.

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_1m_2)+-C(m_2m_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.

Line 89 ⟶ 70:

== 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:

* <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.

Line 190 ⟶ 171:

{{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=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=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 ==