Expander code: Difference between revisions

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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>.

Let <math>\mathcal{S}</math> be an error-correcting code of block length <math>d</math>. The <b><i>expander code</bi></ib> <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>