On Haroutunian’s Exponent for Parallel Channels and an Application to Fixed-Delay Codes Without Feedback

The Haroutunian exponent arises in the study of channel reliability functions for both block coding with feedback and fixed-delay coding without feedback. For asymmetric channels, such as the Z-channel, the Haroutunian exponent is strictly larger than the sphere-packing exponent. The spherepacking exponent is believed to be an upper bound for the reliability function in the two aforementioned communication problems, but in attempting to prove this, one gets stuck at the Haroutunian exponent because of entanglements between the channel behavior and the input distribution. In this paper, we present a characteristic of the Haroutunian exponent that differentiates it from the random coding and sphere-packing exponents. We consider the parallel channel, the repeated use of the original discrete memoryless channel independently some number of times. It is well known that the capacity of the parallel channel is L times the capacity of the original channel, and the random coding and sphere-packing exponents of the L-use parallel channel decompose into L times the exponents of the original channel. The main result of this paper is to show that the (appropriately normalized) Haroutunian exponent of the parallel channel asymptotically decomposes to the sphere-packing exponent of the original channel, as opposed to the Haroutunian exponent of the original channel. This fact is then used to prove two results. First, an upper bound for the reliability function for fixed blocklength coding with delayed feedback is proved. This upper bound converges to the sphere-packing exponent as the delay in the feedback path tends to infinity. Second, the reliability function for fixed delay coding without feedback is shown to be upper bounded by the sphere-packing exponent.