A Sphere-Packing Error Exponent for Mismatched Decoding

Abstract

We derive a sphere-packing error exponent for coded transmission over discrete memoryless channels with a fixed decoding metric. By studying the error probability of the code over an auxiliary channel, we find a lower bound to the probability of error of mismatched decoding. The bound is shown to decay exponentially for coding rates smaller than a new upper bound to the mismatch capacity. For rates higher than the new upper bound, the error probability is shown to be bounded away from zero. The new upper bound is shown to improve over previous upper bounds to the mismatch capacity.

New Converse Bounds on the Mismatched Reliability Function and the Mismatch Capacity Using an Auxiliary Genie Receiver

“…In particular, one needs to verify that ∀z ∈ Z, D q (P Y |X,Z=z ) 0 by checking that the determinants of the 2 |X | − 1 minors of each of these |Z| matrices are all non-negative [31]. This is in contrast to the calculation of C q (W ) and similarly several other previous bounds (e.g., those of [23], [24]), which require to solve a certain minimization problem, such as the minimization in (9), in order to determine whether a two-outputs channel belongs to the set W q (P X ) (see Section IX).…”

“…Note that C q (W ) is quite difficult to compute, since in order to determine whether a two-outputs channel P Y Z|X belongs to the set W q (P X ) or not, one needs to solve the minimization problem in (12). A similar problem arises with the computation of many of the previous bounds (e.g., those of [23], [24]). For this reason, in the next section we present a few looser bounds that are easier to compute.…”

“…A further improved bound was derived in [24] where the set of channels is given by M max (q, P X ) P Y Z|X : min…”

“…In a later work, [23], the class of two-receivers (broadcast) channels was enlarged to include channels satisfying that if the Z-receiver makes an error, then with high probability (approaching 1) so does the Y -receiver. An improved bound was derived in [24], [25], which further enlarged the class of channels. The above mentioned bounds are described in detail in this paper, but most of them are quite complicated to compute in the sense that it is required to solve an optimization problem in order to determine whether a certain channel belongs to the set or not.…”

“…For a wide class of channel-metric pairs, this bound was shown to be tight at R = 0 + . In [24], a new upper bound on the reliability function with mismatched decoding was derived for all rates up to the aforementioned bound of [24] on the mismatch capacity.…”