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The pioneering work of Shannon provides fundamental bounds on the rate limitations of communicating information reliably over noisy channels (the channel coding problem), as well as the compressibility of data subject to distortion constraints (the lossy source coding problem). However, Shannon's theory is nonconstructive in that it only establishes the existence of coding schemes that can achieve the fundamental bounds but provides neither concrete codes nor computationally efficient algorithms. In the case of channel coding, the past two decades have witnessed dramatic advances in practical constructions and algorithms, including the invention of turbo codes and the surge of interest in low-density parity check (LDPC) codes. Both these classes of codes are based on sparse graphs and yield excellent error-correction performance when decoded using computationally efficient methods such as the message-passing sum-product algorithm. Moreover, their performance limits are well characterized, at least in the asymptotic limit of large block lengths, via the density evolution method.
Date of Publication: Sept. 2007