Performance of polar codes for quantum and private classical communication

We analyze the practical performance of quantum polar codes, by computing rigorous bounds on block error probability and by numerically simulating them. We evaluate our bounds for quantum erasure channels with coding block lengths between 2^10 and 2^20, and we report the results of simulations for quantum erasure channels, quantum depolarizing channels, and "BB84" channels with coding block lengths up to N = 1024. For quantum erasure channels, we observe that high quantum data rates can be achieved for block error rates less than 10^(-4) and that somewhat lower quantum data rates can be achieved for quantum depolarizing and BB84 channels. Our results here also serve as bounds for and simulations of private classical data transmission over these channels, essentially due to Renes' duality bounds for privacy amplification and classical data transmission of complementary observables. Future work might be able to improve upon our numerical results for quantum depolarizing and BB84 channels by employing a polar coding rule other than the heuristic used here.Comment: 8 pages, 6 figures, submission to the 50th Annual Allerton Conference on Communication, Control, and Computing 201

Windowed Decoding of Protograph-based LDPC Convolutional Codes over Erasure Channels

We consider a windowed decoding scheme for LDPC convolutional codes that is based on the belief-propagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed decoder. We will consider the performance of these ensembles and codes over erasure channels with and without memory. We show that the structure of LDPC convolutional code ensembles is suitable to obtain performance close to the theoretical limits over the memoryless erasure channel, both for the BP decoder and windowed decoding. However, the same structure imposes limitations on the performance over erasure channels with memory.Comment: 18 pages, 9 figures, accepted for publication in the IEEE Transactions on Information Theor

Serially Concatenated Luby Transform Coding and Bit-Interleaved Coded Modulation Using Iterative Decoding for the Wireless Internet

In Bit-Interleaved Coded Modulation (BICM) the coding and modulation schemes were jointly optimized for the sake of attaining the best possible performance when communicating over fading wireless communication channels. The iterative decoding scheme of BICM (BICM-ID) invoking an appropriate bit-to-symbol mapping strategy enhances its achievable performance in both AWGN and Rayleigh channels. BICM-ID may be conveniently combined with Luby Transform (LT) codes, which were designed for handling packetized wireless Internet data traffic in erasure channels without retransmitting the corrupted packets. By jointly designing a serially concatenated LT-BICM-ID code, an infinitesimally low Bit Error Rate (BER) is achieved for Signal to Noise Ratios (SNR) in excess of 7.5dB over wireless Internet type erasure channels contaminated by AWGN

A Systematic Approach to Incremental Redundancy over Erasure Channels

As sensing and instrumentation play an increasingly important role in systems controlled over wired and wireless networks, the need to better understand delay-sensitive communication becomes a prime issue. Along these lines, this article studies the operation of data links that employ incremental redundancy as a practical means to protect information from the effects of unreliable channels. Specifically, this work extends a powerful methodology termed sequential differential optimization to choose near-optimal block sizes for hybrid ARQ over erasure channels. In doing so, an interesting connection between random coding and well-known constants in number theory is established. Furthermore, results show that the impact of the coding strategy adopted and the propensity of the channel to erase symbols naturally decouple when analyzing throughput. Overall, block size selection is motivated by normal approximations on the probability of decoding success at every stage of the incremental transmission process. This novel perspective, which rigorously bridges hybrid ARQ and coding, offers a pragmatic means to select code rates and blocklengths for incremental redundancy.Comment: 7 pages, 2 figures; A shorter version of this article will appear in the proceedings of ISIT 201

Secret Communication over Broadcast Erasure Channels with State-feedback

We consider a 1-to-$K$ communication scenario, where a source transmits private messages to $K$ receivers through a broadcast erasure channel, and the receivers feed back strictly causally and publicly their channel states after each transmission. We explore the achievable rate region when we require that the message to each receiver remains secret - in the information theoretical sense - from all the other receivers. We characterize the capacity of secure communication in all the cases where the capacity of the 1-to-$K$ communication scenario without the requirement of security is known. As a special case, we characterize the secret-message capacity of a single receiver point-to-point erasure channel with public state-feedback in the presence of a passive eavesdropper. We find that in all cases where we have an exact characterization, we can achieve the capacity by using linear complexity two-phase schemes: in the first phase we create appropriate secret keys, and in the second phase we use them to encrypt each message. We find that the amount of key we need is smaller than the size of the message, and equal to the amount of encrypted message the potential eavesdroppers jointly collect. Moreover, we prove that a dishonest receiver that provides deceptive feedback cannot diminish the rate experienced by the honest receivers. We also develop a converse proof which reflects the two-phase structure of our achievability scheme. As a side result, our technique leads to a new outer bound proof for the non-secure communication problem