Minimum distance of error correcting codes versus encoding complexity, symmetry, and pseudorandomness

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (leaves 207-214).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.We study the minimum distance of binary error correcting codes from the following perspectives: * The problem of deriving bounds on the minimum distance of a code given constraints on the computational complexity of its encoder. * The minimum distance of linear codes that are symmetric in the sense of being invariant under the action of a group on the bits of the codewords. * The derandomization capabilities of probability measures on the Hamming cube based on binary linear codes with good distance properties, and their variations. Highlights of our results include: * A general theorem that asserts that if the encoder uses linear time and sub-linear memory in the general binary branching program model, then the minimum distance of the code cannot grow linearly with the block length when the rate is nonvanishing. * New upper bounds on the minimum distance of various types of Turbo-like codes. * The first ensemble of asymptotically good Turbo like codes. We prove that depth-three serially concatenated Turbo codes can be asymptotically good. * The first ensemble of asymptotically good codes that are ideals in the group algebra of a group. We argue that, for infinitely many block lengths, a random ideal in the group algebra of the dihedral group is an asymptotically good rate half code with a high probability. * An explicit rate-half code whose codewords are in one-to-one correspondence with special hyperelliptic curves over a finite field of prime order where the number of zeros of a codeword corresponds to the number of rational points.(cont.) * A sharp O(k-1/2) upper bound on the probability that a random binary string generated according to a k-wise independent probability measure has any given weight. * An assertion saying that any sufficiently log-wise independent probability measure looks random to all polynomially small read-once DNF formulas. * An elaborate study of the problem of derandomizability of AC₀ by any sufficiently polylog-wise independent probability measure. * An elaborate study of the problem of approximability of high-degree parity functions on binary linear codes by low-degree polynomials with coefficients in fields of odd characteristics.by Louay M.J. Bazzi.Ph.D

Design of tch-type sequences for communications

This thesis deals with the design of a class of cyclic codes inspired by TCH codewords. Since TCH codes are linked to finite fields the fundamental concepts and facts about abstract algebra, namely group theory and number theory, constitute the first part of the thesis. By exploring group geometric properties and identifying an equivalence between some operations on codes and the symmetries of the dihedral group we were able to simplify the generation of codewords thus saving on the necessary number of computations. Moreover, we also presented an algebraic method to obtain binary generalized TCH codewords of length N = 2k, k = 1,2, . . . , 16. By exploring Zech logarithm’s properties as well as a group theoretic isomorphism we developed a method that is both faster and less complex than what was proposed before. In addition, it is valid for all relevant cases relating the codeword length N and not only those resulting from N = p

Nielsen equivalence in Coxeter groups; and a geometric approach to group equivariant machine learning

In this thesis we study two main threads. In Part I, we initiate the study of Nielsen equivalence in Coxeter groups—the classification of finite generating sets up to a natural action of the automorphism group of a free group. We explore different Nielsen equivalence invariants and adapt the method of Lustig and Moriah [79] to the Coxeter case. We also adapt the completion sequences of Dani and Levcovitz [31] to give a method of testing when generating sets of right-angled Coxeter groups are Nielsen equivalent. Coxeter systems have a distinguished set of elements, called the reflections, from which generating sets can be drawn. We study generating sets of reflections separately. In this case, the natural notion of equivalence is generated by partial conjugations of one generator by another. This arises naturally for Weyl groups in the context of cluster algebras via quiver mutations [6]. We study this mutation equivalence for Weyl groups, and reflection equivalence for arbitrary Coxeter systems. In the latter case we leverage hyperplane arrangements in the Davis complex associated to a Coxeter system to give geometric criteria from when a set of reflections generates and a test for when generating sets of reflections are reflection equivalent. In Part II, we discuss the other main topic of the thesis is group equivariant machine learning, based on joint work with Aslan and Platt [3]. We propose a novel approach to defining machine learning algorithms for problems which are equivariant with respect to some discrete group action. Our approach involves pre-processing the input data from a learning algorithm by projecting it onto a fundamental domain for the group action. We give explicit and efficient algorithms for computing this projection. We test our approach on three example learning problems, and demonstrate improvements in accuracy over other methods in the literature

Optimization and Applications of Modern Wireless Networks and Symmetry

Due to the future demands of wireless communications, this book focuses on channel coding, multi-access, network protocol, and the related techniques for IoT/5G. Channel coding is widely used to enhance reliability and spectral efficiency. In particular, low-density parity check (LDPC) codes and polar codes are optimized for next wireless standard. Moreover, advanced network protocol is developed to improve wireless throughput. This invokes a great deal of attention on modern communications

Categorical Quantum Dynamics

We use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Furthermore, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin-type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.Comment: Thesis submitted for the degree of Doctor of Philosophy, Oxford University, Michaelmas Term 2016 (273 pages

LieDetect: Detection of representation orbits of compact Lie groups from point clouds

We suggest a new algorithm to estimate representations of compact Lie groups from finite samples of their orbits. Different from other reported techniques, our method allows the retrieval of the precise representation type as a direct sum of irreducible representations. Moreover, the knowledge of the representation type permits the reconstruction of its orbit, which is useful to identify the Lie group that generates the action. Our algorithm is general for any compact Lie group, but only instantiations for SO(2), T^d, SU(2) and SO(3) are considered. Theoretical guarantees of robustness in terms of Hausdorff and Wasserstein distances are derived. Our tools are drawn from geometric measure theory, computational geometry, and optimization on matrix manifolds. The algorithm is tested for synthetic data up to dimension 16, as well as real-life applications in image analysis, harmonic analysis, and classical mechanics systems, achieving very accurate results.Comment: 84 pages, 16 figure