Expander code: Difference between revisions

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==Definition==

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

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>-left-[[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 <b><i>expander code</b></i> <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}}</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 |chapter-url=http://dl.acm.org/citation.cfm?id=510003 |title=STOC '02 Proceedings of the thirty-fourth annual ACM symposium on Theory of computing |publisher=ACM |year=2002 |isbn=978-1-58113-495-7 |pages=659–668 |doi=10.1145/509907.510003}}</ref>