Extremes of Error Exponents

Error Exponents of Expander Codes

We show that expander codes attain the capacity of the binary-symmetric channel under iterative decoding. The error probability has a positive exponent for all rates between zero and channel capacity. The decoding complexity grows linearly with code length.

Error exponents of expander codes

Error exponents for entanglement concentration

Consider entanglement concentration schemes that convert n identical copies of a pure state into a maximally entangled state of a desired size with success probability being close to one in the asymptotic limit. We give the distillable entanglement, the number of Bell pairs distilled per copy, as a function of an error exponent, which represents the rate of decrease in failure probability as n ...

Statistical mechanics of error exponents for error-correcting codes

Error exponents characterize the exponential decay, when increasing message length, of the probability of error of many error-correcting codes. To tackle the long-standing problem of computing them exactly, we introduce a general, thermodynamic, formalism that we illustrate with maximum-likelihood decoding of low-density parity-check codes on the binary erasure channel and the binary symmetric ...

Error exponents of typical random codes

We define the error exponent of the typical random code as the long–block limit of the negative normalized expectation of the logarithm of the error probability of the random code, as opposed to the traditional random coding error exponent, which is the limit of the negative normalized logarithm of the expectation of the error probability. For the ensemble of uniformly randomly drawn fixed comp...