Expander code[1]

Expander codes are binary linear codes whose parity check matrices are derived from the adjacency matrix of bipartite expander graphs. In particular, the rows of the parity check matrix correspond to the right nodes of the bipartite graph and the columns correspond to the left nodes. The codespace is equivalent to all subsets of the left nodes in the graph that have an even number of edges going into every right node of the graph. Since the expander graph is only left regular, these codes do not qualify as LDPC codes.

Expander codes are important because they admit efficient encoding and decoding algorithms and are asymptotically good (i.e., their rate and normalized distance are constant). The rate and distance of the expander code depend on specific parameters of the corresponding graph. A (\(n, m, D, \gamma, \alpha\)) bipartite expander graph is defined as a \(D\)-left-regular graph with \(n\) left nodes, and \(m\) right nodes such that for any subset of left nodes \(S\) of size at most \(\gamma n\) the neighborhood \(N(S)\) is at least of size \(\alpha|S|\). Given a (\(n, m, D, \gamma, (1-\epsilon)D\)) expander graph, the corresponding expander code has rate of \(1 - m/n\) and a distance of at least \(2(1-\epsilon)\gamma n\) for any \(\epsilon < 1/2\). Explicit constructions for expander graphs with any ratio \(n/m\) are known where \(D = \text{polylog}(n/m)\), \(\gamma = \Omega(1/D)\) and arbitrary \(\epsilon\) [2].