Symmetric Disjunctive List-Decoding Codes

A binary code is said to be a disjunctive list-decoding $s_L$-code (LD $s_L$-code), $s \ge 2$, $L \ge 1$, if the code is identified by the incidence matrix of a family of finite sets in which the union (or disjunctive sum) of any $s$ sets can cover not more than $L-1$ other sets of the family. In this paper, we consider a similar class of binary codes which are based on a {\em symmetric disjunctive sum} (SDS) of binary symbols. By definition, the symmetric disjunctive sum (SDS) takes values from the ternary alphabet $\{0, 1, *\}$, where the symbol~$*$ denotes "erasure". Namely: SDS is equal to $0$ ($1$) if all its binary symbols are equal to $0$ ($1$), otherwise SDS is equal to~$*$. List decoding codes for symmetric disjunctive sum are said to be {\em symmetric disjunctive list-decoding $s_L$-codes} (SLD $s_L$-codes). In the given paper, we remind some applications of SLD $s_L$-codes which motivate the concept of symmetric disjunctive sum. We refine the known relations between parameters of LD $s_L$-codes and SLD $s_L$-codes. For the ensemble of binary constant-weight codes we develop a random coding method to obtain lower bounds on the rate of these codes. Our lower bounds improve the known random coding bounds obtained up to now using the ensemble with independent symbols of codewords.Comment: 18 pages, 1 figure, 1 table, conference pape

Applications of Coding Theory to Massive Multiple Access and Big Data Problems

The broad theme of this dissertation is design of schemes that admit iterative algorithms with low computational complexity to some new problems arising in massive multiple access and big data. Although bipartite Tanner graphs and low-complexity iterative algorithms such as peeling and message passing decoders are very popular in the channel coding literature they are not as widely used in the respective areas of study and this dissertation serves as an important step in that direction to bridge that gap. The contributions of this dissertation can be categorized into the following three parts. In the first part of this dissertation, a timely and interesting multiple access problem for a massive number of uncoordinated devices is considered wherein the base station is interested only in recovering the list of messages without regard to the identity of the respective sources. A coding scheme with polynomial encoding and decoding complexities is proposed for this problem, the two main features of which are (i) design of a close-to-optimal coding scheme for the T-user Gaussian multiple access channel and (ii) successive interference cancellation decoder. The proposed coding scheme not only improves on the performance of the previously best known coding scheme by ≈ 13 dB but is only ≈ 6 dB away from the random Gaussian coding information rate. In the second part construction-D lattices are constructed where the underlying linear codes are nested binary spatially-coupled low-density parity-check codes (SCLDPC) codes with uniform left and right degrees. It is shown that the proposed lattices achieve the Poltyrev limit under multistage belief propagation decoding. Leveraging this result lattice codes constructed from these lattices are applied to the three user symmetric interference channel. For channel gains within 0.39 dB from the very strong interference regime, the proposed lattice coding scheme with the iterative belief propagation decoder, for target error rates of ≈ 10^-5, is only 2:6 dB away the Shannon limit. The third part focuses on support recovery in compressed sensing and the nonadaptive group testing (GT) problems. Prior to this work, sensing schemes based on left-regular sparse bipartite graphs and iterative recovery algorithms based on peeling decoder were proposed for the above problems. These schemes require O(K logN) and Ω(K logK logN) measurements respectively to recover the sparse signal with high probability (w.h.p), where N, K denote the dimension and sparsity of the signal respectively (K (double backward arrow) N). Also the number of measurements required to recover at least (1 - €) fraction of defective items w.h.p (approximate GT) is shown to be cv€_K logN/K. In this dissertation, instead of the left-regular bipartite graphs, left-and- right regular bipartite graph based sensing schemes are analyzed. It is shown that this design strategy enables to achieve superior and sharper results. For the support recovery problem, the number of measurements is reduced to the optimal lower bound of Ω (K log N/K). Similarly for the approximate GT, proposed scheme only requires c€_K log N/ K measurements. For the probabilistic GT, proposed scheme requires (K logK log vN/ K) measurements which is only log K factor away from the best known lower bound of Ω (K log N/ K). Apart from the asymptotic regime, the proposed schemes also demonstrate significant improvement in the required number of measurements for finite values of K, N

Applications of Coding Theory to Massive Multiple Access and Big Data Problems

The broad theme of this dissertation is design of schemes that admit iterative algorithms with low computational complexity to some new problems arising in massive multiple access and big data. Although bipartite Tanner graphs and low-complexity iterative algorithms such as peeling and message passing decoders are very popular in the channel coding literature they are not as widely used in the respective areas of study and this dissertation serves as an important step in that direction to bridge that gap. The contributions of this dissertation can be categorized into the following three parts. In the first part of this dissertation, a timely and interesting multiple access problem for a massive number of uncoordinated devices is considered wherein the base station is interested only in recovering the list of messages without regard to the identity of the respective sources. A coding scheme with polynomial encoding and decoding complexities is proposed for this problem, the two main features of which are (i) design of a close-to-optimal coding scheme for the T-user Gaussian multiple access channel and (ii) successive interference cancellation decoder. The proposed coding scheme not only improves on the performance of the previously best known coding scheme by ≈ 13 dB but is only ≈ 6 dB away from the random Gaussian coding information rate. In the second part construction-D lattices are constructed where the underlying linear codes are nested binary spatially-coupled low-density parity-check codes (SCLDPC) codes with uniform left and right degrees. It is shown that the proposed lattices achieve the Poltyrev limit under multistage belief propagation decoding. Leveraging this result lattice codes constructed from these lattices are applied to the three user symmetric interference channel. For channel gains within 0.39 dB from the very strong interference regime, the proposed lattice coding scheme with the iterative belief propagation decoder, for target error rates of ≈ 10^-5, is only 2:6 dB away the Shannon limit. The third part focuses on support recovery in compressed sensing and the nonadaptive group testing (GT) problems. Prior to this work, sensing schemes based on left-regular sparse bipartite graphs and iterative recovery algorithms based on peeling decoder were proposed for the above problems. These schemes require O(K logN) and Ω(K logK logN) measurements respectively to recover the sparse signal with high probability (w.h.p), where N, K denote the dimension and sparsity of the signal respectively (K (double backward arrow) N). Also the number of measurements required to recover at least (1 - €) fraction of defective items w.h.p (approximate GT) is shown to be cv€_K logN/K. In this dissertation, instead of the left-regular bipartite graphs, left-and- right regular bipartite graph based sensing schemes are analyzed. It is shown that this design strategy enables to achieve superior and sharper results. For the support recovery problem, the number of measurements is reduced to the optimal lower bound of Ω (K log N/K). Similarly for the approximate GT, proposed scheme only requires c€_K log N/ K measurements. For the probabilistic GT, proposed scheme requires (K logK log vN/ K) measurements which is only log K factor away from the best known lower bound of Ω (K log N/ K). Apart from the asymptotic regime, the proposed schemes also demonstrate significant improvement in the required number of measurements for finite values of K, N

Code design and analysis for multiple access communications

This thesis explores various coding aspects of multiple access communications, mainly for spread spectrum multiaccess(SSMA) communications and collaborative coding multiaccess(CCMA) communications. Both the SSMA and CCMA techniques permit efficient simultaneous transmission by several users sharing a common channel, without subdivision in time or frequency. The general principle behind these two multiaccess schemes is that one can find sets of signals (codes) which can be combined together to form a composite signal; on reception, the individual signals in the set can each be recovered from the composite signal. For the CCMA scheme, the isolation between users is based on the code structure; for the SSMA scheme, on the other hand, the isolation between users is based on the autocorrelation functions(ACFs) and crosscorrelation functions (CCFs) of the code sequences. It is clear that, in either case, the code design is the key to the system design.For the CCMA system with a multiaccess binary adder channel, a class of superimposed codes is analyzed. It is proved that every constant weight code of weight w and maximal correlation λ corresponds to a subclass of disjunctive codes of order T 3, the out-of-phase ACFs and CCFs of the codes are constant and equal to √L. In addition, all codes of the same length are mutually orthogonal.2. Maximal length sequences (m-sequences) over Gaussian integers, suitable for use with QAM modulation, are considered. Two sub-classes of m-sequences with quasi-perfect periodic autocorrelations are obtained. The CCFs between the decimated m-sequences are studied. By applying a simple operation, it is shown that some m-sequences over rational and Gaussian integers can be transformed into perfect sequences with impulsive ACFs.3. Frank codes and Chu codes have perfect periodic ACFs and optimum periodic CCFs. In addition, it is shown that they also have very favourable nonperiodic ACFs; some new results concerning the behaviour of the nonperiodic ACFs are derived. Further, it is proved that the sets of combinedFrank/Chu codes, which contain a larger number of codes than either of the two constituent sets, also have very good periodic CCFs. Based on Frank codes and Chu codes, two interesting classes of real-valued codes with good correlation properties are defined. It is shown that these codes have periodic complementary properties and good periodic and nonperiodic ACF/CCFs.Finally, a hybrid CCMA/SSMA coding scheme is proposed. This new hybrid coding scheme provides a very flexible and powerful multiple accessing capability and allows simple and efficient decoding. Given an SSMA system with K users and a CCMA system with N users, where at most T users are active at any time, then the hybrid system will have K . N users with at most T.K users active at any time. The hybrid CCMA/SSMA coding scheme is superior to the individual CCMA system or SSMA system in terms of information rate, number of users, decoding complexity and external interference rejection capability

Code design and analysis for multiple access communications

This thesis explores various coding aspects of multiple access communications, mainly for spread spectrum multiaccess(SSMA) communications and collaborative coding multiaccess(CCMA) communications. Both the SSMA and CCMA techniques permit efficient simultaneous transmission by several users sharing a common channel, without subdivision in time or frequency. The general principle behind these two multiaccess schemes is that one can find sets of signals (codes) which can be combined together to form a composite signal; on reception, the individual signals in the set can each be recovered from the composite signal. For the CCMA scheme, the isolation between users is based on the code structure; for the SSMA scheme, on the other hand, the isolation between users is based on the autocorrelation functions(ACFs) and crosscorrelation functions (CCFs) of the code sequences. It is clear that, in either case, the code design is the key to the system design.For the CCMA system with a multiaccess binary adder channel, a class of superimposed codes is analyzed. It is proved that every constant weight code of weight w and maximal correlation λ corresponds to a subclass of disjunctive codes of order T 3, the out-of-phase ACFs and CCFs of the codes are constant and equal to √L. In addition, all codes of the same length are mutually orthogonal.2. Maximal length sequences (m-sequences) over Gaussian integers, suitable for use with QAM modulation, are considered. Two sub-classes of m-sequences with quasi-perfect periodic autocorrelations are obtained. The CCFs between the decimated m-sequences are studied. By applying a simple operation, it is shown that some m-sequences over rational and Gaussian integers can be transformed into perfect sequences with impulsive ACFs.3. Frank codes and Chu codes have perfect periodic ACFs and optimum periodic CCFs. In addition, it is shown that they also have very favourable nonperiodic ACFs; some new results concerning the behaviour of the nonperiodic ACFs are derived. Further, it is proved that the sets of combinedFrank/Chu codes, which contain a larger number of codes than either of the two constituent sets, also have very good periodic CCFs. Based on Frank codes and Chu codes, two interesting classes of real-valued codes with good correlation properties are defined. It is shown that these codes have periodic complementary properties and good periodic and nonperiodic ACF/CCFs.Finally, a hybrid CCMA/SSMA coding scheme is proposed. This new hybrid coding scheme provides a very flexible and powerful multiple accessing capability and allows simple and efficient decoding. Given an SSMA system with K users and a CCMA system with N users, where at most T users are active at any time, then the hybrid system will have K . N users with at most T.K users active at any time. The hybrid CCMA/SSMA coding scheme is superior to the individual CCMA system or SSMA system in terms of information rate, number of users, decoding complexity and external interference rejection capability

Near-Optimal Noisy Group Testing via Separate Decoding of Items

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and more. In this paper, we revisit an efficient algorithm for noisy group testing in which each item is decoded separately (Malyutov and Mateev, 1980), and develop novel performance guarantees via an information-theoretic framework for general noise models. For the special cases of no noise and symmetric noise, we find that the asymptotic number of tests required for vanishing error probability is within a factor $\log 2 \approx 0.7$ of the information-theoretic optimum at low sparsity levels, and that with a small fraction of allowed incorrectly decoded items, this guarantee extends to all sublinear sparsity levels. In addition, we provide a converse bound showing that if one tries to move slightly beyond our low-sparsity achievability threshold using separate decoding of items and i.i.d. randomized testing, the average number of items decoded incorrectly approaches that of a trivial decoder.Comment: Submitted to IEEE Journal of Selected Topics in Signal Processin