Expander code: Difference between revisions

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In [[coding theory]], '''expander codes''' form a class of [[Error detection and correction|error-correcting codes]] that are constructed from [[Bipartite graph|bipartite]] [[expander graph]]s.

Along with [[Justesen code]]s, expander codes are of particular interest since they have a constant positive [[Block_codeBlock code#The_rate_RThe rate R|rate]], a constant positive relative [[Block_codeBlock code#The_distance_dThe distance d|distance]], and a constant [[Block_codeBlock code#The_alphabet_The alphabet .CE.A3|alphabet size]].

In fact, the alphabet contains only two elements, so expander codes belong to the class of [[binary code]]s.

Furthermore, expander codes can be both encoded and decoded in time proportional to the block length of the code.

Expander codes are the only known asymptotically good codes which can be both encoded and decoded from a constant fraction of errors in polynomial time.

==Expander codes==

In [[coding theory]], an expander code is a <math>[n,n-m]_2\,</math> [[linear block code]] whose parity check matrix is the adjacency matrix of a bipartite [[expander graph]]. These codes have good relative [[Block_codeBlock code#The_distance_dThe distance d|distance]] <math>2(1-\varepsilon)\gamma\,</math>, where <math>\varepsilon\,</math> and <math>\gamma\,</math> are properties of the expander graph as defined later), [[Block_codeBlock code#The_rate_RThe rate R|rate]] <math>\left(1-\tfrac{m}{n}\right)\,</math>, and decodability (algorithms of running time <math>O(n)\,</math> exist).

==Definition==

Let <math>B</math> be a <math>(c,d)</math>-[[biregular graph]] between a set of <math>n</math> nodes <math>\{v_1,\cdots,v_n\}</math>, called ''variables'', and a set of <math>cn/d</math> nodes <math>\{C_1,\cdots,C_{cn/d}\}</math>, called ''constraints''.

Consider a [[bipartite graph]] <math>G(L,R,E)\,</math>, where <math>L\,</math> and <math>R\,</math> are the vertex sets and <math>E\,</math> is the set of edges connecting vertices in <math>L\,</math> to vertices of <math>R\,</math>. Suppose every vertex in <math>L\,</math> has [[degree (graph theory)|degree]] <math>d\,</math> (the graph is <math>d\,</math>-[[Regular graph|regular]]), <math>|L|=n\,</math>, and <math>|R|=m\,</math>, <math>m < n\,</math>. Then <math>G\,</math> is a <math>(N, M, d, \gamma, \alpha)\,</math> expander graph if every small enough subset <math>S \subset L\,</math>, <math>|S| \leq \gamma n\,</math> has the property that <math>S\,</math> has at least <math>d\alpha|S|\,</math> distinct neighbors in <math>R\,</math>. Note that this holds trivially for <math>\gamma \leq \tfrac{1}{n}\,</math>. When <math>\tfrac{1}{n} < \gamma \leq 1\,</math> and <math>\alpha = 1 - \varepsilon\,</math> for a constant <math>\varepsilon\,</math>, we say that <math>G\,</math> is a lossless expander.

Let <math>b(i,j)</math> be a function designed so that, for each constraint <math>C_i</math>, the variables neighboring <math>C_i</math> are <math>v_{b(i,1)},\cdots,v_{b(i,d)}</math>.

Since <math>G\,</math> is a bipartite graph, we may consider its <math>n \times m\,</math> adjacency matrix. Then the linear code <math>C\,</math> generated by viewing the transpose of this matrix as a parity check matrix is an expander code.

Let <math>\mathcal{S}</math> be an error-correcting code of block length <math>d</math>. The '''''expander code''''' <math>\mathcal{C}(B,\mathcal{S})</math> is the code of block length <math>n</math> whose codewords are the words <math>(x_1,\cdots,x_n)</math> such that, for <math>1\leq i\leq cn/d</math>, <math>(x_{b(i,1)},\cdots,x_{b(i,d)})</math> is a codeword of <math>\mathcal{S}</math>.<ref name="definition">{{cite journal|doi=10.1109/18.556667|title=Expander codes |year=1996 |last1=Sipser |first1=M. |last2=Spielman |first2=D.A. |journal=IEEE Transactions on Information Theory |volume=42 |issue=6 |pages=1710–1722 }}</ref>

It has been shown that nontrivial lossless expander graphs exist. Moreover, we can explicitly construct them.<ref name="lossless">{{cite book |first1=M. |last1=Capalbo |first2=O. |last2=Reingold |first3=S. |last3=Vadhan |first4=A. |last4=Wigderson |chapter=Randomness conductors and constant-degree lossless expanders |chapterurl=http://dl.acm.org/citation.cfm?id=510003 |editor= |title=STOC '02 Proceedings of the thirty-fourth annual ACM symposium on Theory of computing |publisher=ACM |___location= |year=2002 |isbn=978-1-58113-495-7 |pages=659–668 |doi=10.1145/509907.510003}}</ref>

It has been shown that nontrivial lossless expander graphs exist. Moreover, we can explicitly construct them.<ref name="lossless">{{cite book |first1=M. |last1=Capalbo |first2=O. |last2=Reingold |first3=S. |last3=Vadhan |first4=A. |last4=Wigderson |chapter=Randomness conductors and constant-degree lossless expanders |chapterurlchapter-url=http://dl.acm.org/citation.cfm?id=510003 |editor= |title=STOC '02 Proceedings of the thirty-fourth annual ACM symposium on Theory of computing |publisher=ACM |___location= |year=2002 |isbn=978-1-58113-495-7 |pages=659–668 |doi=10.1145/509907.510003|s2cid=1918841 }}</ref>

==Rate==

The rate of <math>C\,</math> is its dimension divided by its block length. In this case, the parity check matrix has size <math>m \times n\,</math>, and hence <math>C\,</math> has dimensionrate at least <math>(n-m)/n = 1 - \tfrac{m}{/n}\,</math>.

==Distance==

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==Encoding==

The encoding time for an expander code is upper bounded by that of a general linear code - <math>O(n^2)\,</math> by matrix multiplication. A result due to Spielman shows that encoding is possible in <math>O(n)\,</math> time.<ref name="spielman">{{cite journal |first=D. |last=Spielman |title=Linear-time encodable and decodable error-correcting codes |journal=IEEE Transactions on Information Theory |volume=42 |issue=6 |pages=1723–31 |year=1996 |doi=10.1109/18.556668 |url=|citeseerx=10.1.1.47.2736 }}</ref>

==Decoding==

Decoding of expander codes is possible in <math>O(n)\,</math> time when <math>\varepsilon < \tfrac{1}{4}\,</math> using the following algorithm.

Let <math>v_i\,</math> be the vertex of <math>L\,</math> that corresponds to the <math>i\,</math>th index in the codewords of <math>C\,</math>. Let <math>y \in \{0,1\}^n\,</math> be a received word, and <math>V(y) = \{v_i |\mid \text{ the } i^{\text{th}} \text{ position of } y \text{ is a } 1\}\,</math>. Let <math>e(i)\,</math> be <math>|\{v \in R |\mid v_i \in N(v) \text{ and } N(v) \cap V(y) \,</math>text{ is even<math>}\}|\,</math>, and <math>o(i)\,</math> be <math>|\{v \in R |\mid v_i \in N(v) \text{ and } N(v) \cap V(y) \,</math>text{ is odd<math>}\}|\,</math>. Then consider the greedy algorithm:

----

'''Input:''' received codewordword <math>y\,</math>.

<code>

initialize y' to y

while there is a v in R adjacent to an odd number of vertices in V(y')

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flip entry i in y'

else

fail</code>

'''Output:''' fail, or modified codeword <math>y'\,</math>.

----

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==Notes==

This article is based on Dr. Venkatesan Guruswami's course notes.<ref>{{cite web |first=V. |last=Guruswami |title=Lecture 13: Expander Codes |date=15 November 2006 |work=CSE 533: Error-Correcting |publisher=University of Washington |url=http://www.cs.washington.edu/education/courses/cse533/06au/lecnotes/lecture13.pdf }}<br/>

{{cite web |first=V. |last=Guruswami |title=Notes 8: Expander Codes and their decoding |date=March 2010 |work=Introduction to Coding Theory |publisher=Carnegie Mellon University |url=httphttps://www.cs.cmu.edu/~venkatg/teaching/codingtheory/notes/notes8.pdf }}<br/>

{{cite journal |first=V. |last=Guruswami |title=Guest column: error-correcting codes and expander graphs |journal=ACM SIGACT News |volume=35 |issue=3 |pages=25–41 |date=September 2004 |doi=10.1145/1027914.1027924 |s2cid=17550280 |url=http://dl.acm.org/citation.cfm?id=1027924|url-access=subscription }}</ref>

==References==