Applications of ordered weights in information transmission

This dissertation is devoted to a study of a class of linear codes related to a particular metric space that generalizes the Hamming space in that the metric function is defined by a partial order on the set of coordinates of the vector. We begin with developing combinatorial and linear-algebraic aspects of linear ordered codes. In particular, we define multivariate rank enumerators for linear codes and show that they form a natural set of invariants in the study of the duality of linear codes. The rank enumerators are further shown to be connected to the shape distributions of linear codes, and enable us to give a simple proof of a MacWilliams-like theorem for the ordered case. We also pursue the connection between linear codes and matroids in the ordered case and show that the rank enumerator can be thought of as an instance of the classical matroid invariant called the Tutte polynomial. Finally, we consider the distributions of support weights of ordered codes and their expression via the rank enumerator. Altogether, these results generalize a group of well-known results for codes in the Hamming space to the ordered case. Extending the research in the first part, we define simple probabilistic channel models that are in a certain sense matched to the ordered distance, and prove several results related to performance of linear codes on such channels. In particular, we define ordered wire-tap channels and establish several results related to the use of linear codes for reliable and secure transmission in such channel models. In the third part of this dissertation we study polar coding schemes for channels with nonbinary input alphabets. We construct a family of linear codes that achieve the capacity of a nonbinary symmetric discrete memoryless channel with input alphabet of size q=2^r, r=2,3,.... A new feature of the coding scheme that arises in the nonbinary case is related to the emergence of several extremal configurations for the polarized data symbols. We establish monotonicity properties of the configurations and use them to show that total transmission rate approaches the symmetric capacity of the channel. We develop these results to include the case of ``controlled polarization'' under which the data symbols polarize to any predefined set of extremal configurations. We also outline an application of this construction to data encoding in video sequences of the MPEG-2 and H.264/MPEG-4 standards

Coding and Decoding Schemes for MSE and Image Transmission

In this work we explore possibilities for coding and decoding tailor-made for mean squared error evaluation of error in contexts such as image transmission. To do so, we introduce a loss function that expresses the overall performance of a coding and decoding scheme for discrete channels and that exchanges the usual goal of minimizing the error probability to that of minimizing the expected loss. In this environment we explore the possibilities of using ordered decoders to create a message-wise unequal error protection (UEP), where the most valuable information is protected by placing in its proximity information words that differ by a small valued error. We give explicit examples, using scale-of-gray images, including small-scale performance analysis and visual simulations for the BSMC.Comment: Submitted to IEEE Transactions on Information Theor

Generalized List Decoding

This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip ($\textsf{XOR}$) channels, erasure channels, $\textsf{AND}$ ($Z$-) channels, $\textsf{OR}$ channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) $(L-1)$-list decodable codes (where the list size $L$ is a universal constant) exist for such channels. Our criterion asserts that: "For any given general adversarial channel, it is possible to construct positive rate $(L-1)$-list decodable codes if and only if the set of completely positive tensors of order-$L$ with admissible marginals is not entirely contained in the order-$L$ confusability set associated to the channel." The sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by 1. extracting equicoupled subcodes (generalization of equidistant code) from any large code sequence using hypergraph Ramsey's theorem, and 2. significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone. In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which be may of independent interest. Other results include 1. List decoding capacity with asymptotically large $L$ for general adversarial channels; 2. A tight list size bound for most constant composition codes (generalization of constant weight codes); 3. Rederivation and demystification of Blinovsky's [Bli86] characterization of the list decoding Plotkin points (threshold at which large codes are impossible); 4. Evaluation of general bounds ([WBBJ]) for unique decoding in the error correction code setting

Asymmetric Lee Distance Codes for DNA-Based Storage

We consider a new family of codes, termed asymmetric Lee distance codes, that arise in the design and implementation of DNA-based storage systems and systems with parallel string transmission protocols. The codewords are defined over a quaternary alphabet, although the results carry over to other alphabet sizes; furthermore, symbol confusability is dictated by their underlying binary representation. Our contributions are two-fold. First, we demonstrate that the new distance represents a linear combination of the Lee and Hamming distance and derive upper bounds on the size of the codes under this metric based on linear programming techniques. Second, we propose a number of code constructions which imply lower bounds