Recursive quantum convolutional encoders are catastrophic: A simple proof

Poulin, Tillich, and Ollivier discovered an important separation between the classical and quantum theories of convolutional coding, by proving that a quantum convolutional encoder cannot be both non-catastrophic and recursive. Non-catastrophicity is desirable so that an iterative decoding algorithm converges when decoding a quantum turbo code whose constituents are quantum convolutional codes, and recursiveness is as well so that a quantum turbo code has a minimum distance growing nearly linearly with the length of the code, respectively. Their proof of the aforementioned theorem was admittedly "rather involved," and as such, it has been desirable since their result to find a simpler proof. In this paper, we furnish a proof that is arguably simpler. Our approach is group-theoretic---we show that the subgroup of memory states that are part of a zero physical-weight cycle of a quantum convolutional encoder is equivalent to the centralizer of its "finite-memory" subgroup (the subgroup of memory states which eventually reach the identity memory state by identity operator inputs for the information qubits and identity or Pauli-Z operator inputs for the ancilla qubits). After proving that this symmetry holds for any quantum convolutional encoder, it easily follows that an encoder is non-recursive if it is non-catastrophic. Our proof also illuminates why this no-go theorem does not apply to entanglement-assisted quantum convolutional encoders---the introduction of shared entanglement as a resource allows the above symmetry to be broken.Comment: 15 pages, 1 figure. v2: accepted into IEEE Transactions on Information Theory with minor modifications. arXiv admin note: text overlap with arXiv:1105.064

Digital communications techniques Interim report, 15 Sep. 1969 - 15 Feb. 1970

Convolutional codes and recursive signal processing for digital communication

Study and simulation of low rate video coding schemes

The semiannual report is included. Topics covered include communication, information science, data compression, remote sensing, color mapped images, robust coding scheme for packet video, recursively indexed differential pulse code modulation, image compression technique for use on token ring networks, and joint source/channel coder design

An Iterative Joint Linear-Programming Decoding of LDPC Codes and Finite-State Channels

In this paper, we introduce an efficient iterative solver for the joint linear-programming (LP) decoding of low-density parity-check (LDPC) codes and finite-state channels (FSCs). In particular, we extend the approach of iterative approximate LP decoding, proposed by Vontobel and Koetter and explored by Burshtein, to this problem. By taking advantage of the dual-domain structure of the joint decoding LP, we obtain a convergent iterative algorithm for joint LP decoding whose structure is similar to BCJR-based turbo equalization (TE). The result is a joint iterative decoder whose complexity is similar to TE but whose performance is similar to joint LP decoding. The main advantage of this decoder is that it appears to provide the predictability of joint LP decoding and superior performance with the computational complexity of TE.Comment: To appear in Proc. IEEE ICC 2011, Kyoto, Japan, June 5-9, 201

Investigation of Different Constituent Encoders in a Turbo-code Scheme for Reduced Decoder Complexity

A large number of papers have been published attempting to give some analytical basis for the performance of Turbo-codes. It has been shown that performance improves with increased interleaver length. Also procedures have been given to pick the best constituent recursive systematic convolutional codes (RSCC's). However testing by computer simulation is still required to verify these results. This thesis begins by describing the encoding and decoding schemes used. Next simulation results on several memory 4 RSCC's are shown. It is found that the best BER performance at low E(sub b)/N(sub o) is not given by the RSCC's that were found using the analytic techniques given so far. Next the results are given from simulations using a smaller memory RSCC for one of the constituent encoders. Significant reduction in decoding complexity is obtained with minimal loss in performance. Simulation results are then given for a rate 1/3 Turbo-code with the result that this code performed as well as a rate 1/2 Turbo-code as measured by the distance from their respective Shannon limits. Finally the results of simulations where an inaccurate noise variance measurement was used are given. From this it was observed that Turbo-decoding is fairly stable with regard to noise variance measurement

Turbo Decoding and Detection for Wireless Applications

A historical perspective of turbo coding and turbo transceivers inspired by the generic turbo principles is provided, as it evolved from Shannon’s visionary predictions. More specifically, we commence by discussing the turbo principles, which have been shown to be capable of performing close to Shannon’s capacity limit. We continue by reviewing the classic maximum a posteriori probability decoder. These discussions are followed by studying the effect of a range of system parameters in a systematic fashion, in order to gauge their performance ramifications. In the second part of this treatise, we focus our attention on the family of iterative receivers designed for wireless communication systems, which were partly inspired by the invention of turbo codes. More specifically, the family of iteratively detected joint coding and modulation schemes, turbo equalization, concatenated spacetime and channel coding arrangements, as well as multi-user detection and three-stage multimedia systems are highlighted

Signal Codes

Motivated by signal processing, we present a new class of channel codes, called signal codes, for continuous-alphabet channels. Signal codes are lattice codes whose encoding is done by convolving an integer information sequence with a fixed filter pattern. Decoding is based on the bidirectional sequential stack decoder, which can be implemented efficiently using the heap data structure. Error analysis and simulation results indicate that signal codes can achieve low error rate at approximately 1dB from channel capacity.Comment: Submitted to IEEE Transactions on Information Theor