Error-and-Erasure Decoding for Block Codes with Feedback

Shannon showed in [18] that the capacity of discrete memoryless channels (DMCs) does not increase even when a noiseless and delay free feedback link exists from receiver to transmitter. Moreover on symmetric DMCs the sphere packing bound is an upper bound for the error exponent of fixed-length block-codes, as shown by Dobrushin1 in [5]. Thus relaxations like error-and-erasure decoding or variable length coding are needed for feedback to increase the error exponent of block-codes at rates larger than the critical rate on symmetric DMCs. In this work investigate one such relaxation, namely errors-and-erasures decoding and find inner and outer bounds to the optimal error exponent erasure exponent trade off. Before doing that let us briefly summarize the previous work on certain related problems and relate those results to the ones we present in this work. Burnashev, [1], considered variable-length block-codes on DMCs with feedback, instead of fixed-length ones and obtained the exact expression for the error exponent at all rates.2 Later Yamamoto and Itoh, [22], suggested a coding scheme which achieves the best error exponent for variable-length block-codes with feedback by using a fixed-length block-code with error-and-erasure decoding, repetitively until decoding without erasures occurs. One can reinterpret [1] and [22] to get the error exponent of fixed-length block-codes with error-and-erasure decoding and feedback at all rates below capacity, when erasure probability is decaying sub-exponentially with block length. However the average transmission time is only a first order measure, for analyzing the benefits of error-and-erasure decoding for block-codes. In other words as long as the erasure probability is vanishing, with increasing block length, average transmission time will essentially be equal to the block length of the fixed-length block-code. Thus with an analysis like the one in [22] the cost of retransmissions are ignored as long as the erasure probability goes to zero with increasing block length. On the other hand in a communication system with multiple layers, retransmissions can have costs beyond their effect on average transmission time, i.e. we might need to include bounds on higher moments of the transmission time in our model. The knowledge of the trade-off between the exponential decay rates of error probability and erasure probability will not only help us analyze those models but also give us an upper hand in the design of multi layer communication systems in general . A separate stream of research focused on the benefits of error-and-erasure decoding for block-codes on DMC’s without feedback. First Forney, [7], considered error-and-erasure decoding without feedback and obtained an achievable trade-off between the exponents of error and erasure probabilities. Then Csiszár and Körner, [4] achieved