Expander code: Difference between revisions

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==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 code#The 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 code#The 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>-regular[[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''.

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>.

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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=https://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==