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The error exponent of the binary symmetric channel for asymmetric random codes

In this paper, we consider asymmetric binary codes from Shannon's random code ensemble and also from the random linear code ensemble (LCE), used over the the binary symmetric channel (BSC). The asymmetry is in the sense that the codeword symbols are not necessarily chosen in an equiprobable manner. One possible motivation for such asymmetry is mismatch, where the user uses asymmetric codes over the BSC, leading to a degradation in performance. Accordingly, we derive the distance distribution and the error exponents of a typical random code (TRC) from the RCE, and of a typical linear code (TLC) from the LCE. The derivation is based on a fine large-deviation analysis of some distance enumerators, contrary to the usual bounding technique by Gallager. Later, we propose a “time-varying” BSC model, in which the crossover probability of the BSC is time-dependent, and use our results for providing a lower bound on the error exponent of this channel model.