Noisy-channel coding theorem: Difference between revisions

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The channel capacity <math>C</math> can be calculated from the physical properties of a channel; for a band-limited channel with Gaussian noise, using the [[Shannon–Hartley theorem]].

Simple schemes such as "send the message 3 times and use a best 2 out of 3 voting scheme if the copies differ" are inefficient error-correction methods, unable to asymptotically guarantee that a block of data can be communicated free of error. Advanced techniques such as [[Reed–Solomon code]]s and, more recently, [[low-density parity-check code|low-density parity-check]] (LDPC) codes and [[turbo code]]s, come much closer to reaching the theoretical Shannon limit, but at a cost of high computational complexity. Using these highly efficient codes and with the computing power in today's [[digital signal processors]], it is now possible to reach very close to the Shannon limit. In fact, it was shown that LDPC codes can reach within 0.0045&nbsp;dB of the Shannon limit (for binary [[Additiveadditive white Gaussian noise]] (AWGN) channels, with very long block lengths).<ref>[[Sae-Young Chung]], [[G. David Forney, Jr.]], [[Thomas J. Richardson]], and [[Rüdiger Urbanke]], "[http://www.josephboutros.org/ldpc_vs_turbo/ldpc_Chung_CLfeb01.pdf On the Design of Low-Density Parity-Check Codes within 0.0045 dB of the Shannon Limit]", ''[[IEEE Communications Letters]]'', 5: 58-60, Feb. 2001. ISSN 1089-7798</ref>

== Mathematical statement ==