Judy L. Walker

Efficient Traitor Tracing Algorithms Using List Decoding

We use powerful new techniques for list decoding errorcorrecting codes to efficiently trace traitors. Although much work has focusedon constructing traceability schemes, the complexity of the tracing algorithm has receivedlittle attention. Because the TA tracing algorithm has a runtime of O(N) in general, where N is the number of users, it is inefficient for large populations.We produce schemes for which the TA algorithm is very fast. The IPP tracing algorithm, though less efficient, can list all coalitions capable of constructing a given pirate. We give evidence that when using an algebraic structure, the ability to trace with the IPP algorithm implies the ability to trace with the TA algorithm. We also construct schemes with an algorithm that finds all possible traitor coalitions faster than the IPP algorithm. Finally, we suggest uses for other decoding techniques in the presence of additional information about traitor behavior.

Applications of list decoding to tracing traitors

We apply results from algebraic coding theory to solve problems in cryptography, by using recent results on list decoding of error-correcting codes to efficiently find traitors who collude to create pirates. We produce schemes for which the traceability (TA) traitor tracing algorithm is very fast. We compare the TA and identifiable parent property (IPP) traitor tracing algorithms, and give evidence that when using an algebraic structure, the ability to trace traitors with the IPP algorithm implies the ability to trace with the TA algorithm. We also demonstrate that list decoding techniques can be used to find all possible pirate coalitions. Finally, we raise some related open questions about linear codes, and suggest uses for other decoding techniques in the presence of additional information about traitor behavior.

Average min-sum decoding of LDPC codes

Simulations have shown that the outputs of min-sum (MS) decoding generally behave in one of two ways: either the output vector eventually stabilizes at a codeword or it eventually cycles through a finite set of vectors that may include both codewords and non-codewords. The latter behavior has significantly contributed to the difficulty in studying the performance of this decoder. To overcome this problem, a new decoder, average min-sum (AMS), is proposed; this decoder outputs the average of the MS output vectors over a finite set of iterations. Simulations comparing MS, AMS, linear programming (LP) decoding, and maximum likelihood (ML) decoding are presented, illustrating the relative performances of each of these decoders. In general, MS and AMS have comparable word error rates; however, in the simulation of a code with large block length, AMS has a significantly lower bit error rate. Finally, AMS pseudocodewords are introduced and their relationship to graph cover and LP pseudocodewords is explored, with particular focus on the AMS pseudocodewords of regular LDPC codes and cycle codes.

Pseudo-codewords of cycle codes via zeta functions

Cycle codes are a special case of low-density parity-check (LDPC) codes and as such can be decoded using an iterative message-passing decoding algorithm on the associated Tanner graph. The existence of pseudo-codewords is known to cause the decoding algorithm to fail in certain instances. In this paper, we draw a connection between pseudo-codewords of cycle codes and the so-called edge zeta function of the associated normal graph and show how the Newton polytope of the zeta function equals the fundamental cone of the code, which plays a crucial role in characterizing the performance of iterative decoding algorithms.

Nonbinary quantum error-correcting codes from algebraic curves

We give a new exposition and proof of a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes with various lengths, dimensions, and minimum distances from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia-Stichtenoth tower of function fields which are constructible in polynomial time.

Combinatorial neural codes from a mathematical coding theory perspective

Shannons seminal 1948 work gave rise to two distinct areas of research: information theory and mathematical coding theory. While information theory has had a strong influence on theoretical neuroscience, ideas from mathematical coding theory have received considerably less attention. Here we take a new look at combinatorial neural codes from a mathematical coding theory perspective, examining the error correction capabilities of familiar receptive field codes (RF codes). We find, perhaps surprisingly, that the high levels of redundancy present in these codes do not support accurate error correction, although the error-correcting performance of receptive field codes catches up to that of random comparison codes when a small tolerance to error is introduced. However, receptive field codes are good at reflecting distances between represented stimuli, while the random comparison codes are not. We suggest that a compromise in error-correcting capability may be a necessary price to pay for a neural code whose structure serves not only error correction, but must also reflect relationships between stimuli.

Analysis of Connections Between Pseudocodewords

The role of pseudocodewords in causing non-codeword outputs in linear programming decoding, graph cover decoding, and iterative message-passing decoding is investigated. The three main types of pseudocodewords in the literature-linear programming pseudocodewords, graph cover pseudocodewords, and computation tree pseudocodewords-are reviewed and connections between them are explored. Some discrepancies in the literature on minimal and irreducible pseudocodewords are highlighted and clarified, and the minimal degree cover necessary to realize a pseudocodeword is found. Additionally, some conditions for the existence of connected realizations of graph cover pseudocodewords are given. This allows for further analysis of when graph cover pseudocodewords induce computation tree pseudocodewords. Finally, an example is offered that shows that existing theories on the distinction between graph cover pseudocodewords and computation tree pseudocodewords are incomplete.

Euclidean weights of codes from elliptic curves over rings

We construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools, notably an estimate for certain exponential sums and some results on canonical lifts of elliptic curves. These results may be of independent interest.

Algebraic Geometric Codes over Rings

Abstract The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite fields since the early 1980s. Recently, there has been an increased interest in the study of linear codes over finite rings. In this paper, we combine these two approaches to coding theory by introducing the study of algebraic geometric codes over rings. In addition to defining these new codes, we prove several results about their properties.

LDPC codes from voltage graphs

Several well-known structure-based constructions of LDPC codes, for example codes based on permutation and circulant matrices and in particular, quasi-cyclic LDPC codes, can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph theory can be used to give simple proofs of many known properties of these codes. In addition, the notion of Abelian-inevitable cycle is introduced and the subgraphs giving rise to these cycles are classified. We also indicate how, by using more sophisticated voltage assignments, new classes of good LDPC codes may be obtained.