A linear construction for certain Kerdock and Preparata codes

The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes are shown to be linear over \ZZ_4, the integers $\bmod~4$. The Kerdock and Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over \ZZ_4. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in general are not, nor is the Golay code.Comment: 5 page

Quantum error-correcting codes associated with graphs

We present a construction scheme for quantum error correcting codes. The basic ingredients are a graph and a finite abelian group, from which the code can explicitly be obtained. We prove necessary and sufficient conditions for the graph such that the resulting code corrects a certain number of errors. This allows a simple verification of the 1-error correcting property of fivefold codes in any dimension. As new examples we construct a large class of codes saturating the singleton bound, as well as a tenfold code detecting 3 errors.Comment: 8 pages revtex, 5 figure

Scalar--Flat Lorentzian Einstein--Weyl Spaces

We find all three-dimensional Einstein--Weyl spaces with the vanishing scalar curvatureComment: 4 page

Embedded Rank Distance Codes for ISI channels

Designs for transmit alphabet constrained space-time codes naturally lead to questions about the design of rank distance codes. Recently, diversity embedded multi-level space-time codes for flat fading channels have been designed from sets of binary matrices with rank distance guarantees over the binary field by mapping them onto QAM and PSK constellations. In this paper we demonstrate that diversity embedded space-time codes for fading Inter-Symbol Interference (ISI) channels can be designed with provable rank distance guarantees. As a corollary we obtain an asymptotic characterization of the fixed transmit alphabet rate-diversity trade-off for multiple antenna fading ISI channels. The key idea is to construct and analyze properties of binary matrices with a particular structure induced by ISI channels.Comment: Submitted to IEEE Transactions on Information Theor

List decoding of noisy Reed-Muller-like codes

First- and second-order Reed-Muller (RM(1) and RM(2), respectively) codes are two fundamental error-correcting codes which arise in communication as well as in probabilistically-checkable proofs and learning. In this paper, we take the first steps toward extending the quick randomized decoding tools of RM(1) into the realm of quadratic binary and, equivalently, Z_4 codes. Our main algorithmic result is an extension of the RM(1) techniques from Goldreich-Levin and Kushilevitz-Mansour algorithms to the Hankel code, a code between RM(1) and RM(2). That is, given signal s of length N, we find a list that is a superset of all Hankel codewords phi with dot product to s at least (1/sqrt(k)) times the norm of s, in time polynomial in k and log(N). We also give a new and simple formulation of a known Kerdock code as a subcode of the Hankel code. As a corollary, we can list-decode Kerdock, too. Also, we get a quick algorithm for finding a sparse Kerdock approximation. That is, for k small compared with 1/sqrt{N} and for epsilon > 0, we find, in time polynomial in (k log(N)/epsilon), a k-Kerdock-term approximation s~ to s with Euclidean error at most the factor (1+epsilon+O(k^2/sqrt{N})) times that of the best such approximation

Compressive Classification

This paper derives fundamental limits associated with compressive classification of Gaussian mixture source models. In particular, we offer an asymptotic characterization of the behavior of the (upper bound to the) misclassification probability associated with the optimal Maximum-A-Posteriori (MAP) classifier that depends on quantities that are dual to the concepts of diversity gain and coding gain in multi-antenna communications. The diversity, which is shown to determine the rate at which the probability of misclassification decays in the low noise regime, is shown to depend on the geometry of the source, the geometry of the measurement system and their interplay. The measurement gain, which represents the counterpart of the coding gain, is also shown to depend on geometrical quantities. It is argued that the diversity order and the measurement gain also offer an optimization criterion to perform dictionary learning for compressive classification applications.Comment: 5 pages, 3 figures, submitted to the 2013 IEEE International Symposium on Information Theory (ISIT 2013

Mismatch in the Classification of Linear Subspaces: Sufficient Conditions for Reliable Classification

This paper considers the classification of linear subspaces with mismatched classifiers. In particular, we assume a model where one observes signals in the presence of isotropic Gaussian noise and the distribution of the signals conditioned on a given class is Gaussian with a zero mean and a low-rank covariance matrix. We also assume that the classifier knows only a mismatched version of the parameters of input distribution in lieu of the true parameters. By constructing an asymptotic low-noise expansion of an upper bound to the error probability of such a mismatched classifier, we provide sufficient conditions for reliable classification in the low-noise regime that are able to sharply predict the absence of a classification error floor. Such conditions are a function of the geometry of the true signal distribution, the geometry of the mismatched signal distributions as well as the interplay between such geometries, namely, the principal angles and the overlap between the true and the mismatched signal subspaces. Numerical results demonstrate that our conditions for reliable classification can sharply predict the behavior of a mismatched classifier both with synthetic data and in a motion segmentation and a hand-written digit classification applications.Comment: 17 pages, 7 figures, submitted to IEEE Transactions on Signal Processin