Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound

We introduce synchronization strings as a novel way of efficiently dealing with synchronization errors, i.e., insertions and deletions. Synchronization errors are strictly more general and much harder to deal with than commonly considered half-errors, i.e., symbol corruptions and erasures. For every $\epsilon >0$, synchronization strings allow to index a sequence with an $\epsilon^{-O(1)}$ size alphabet such that one can efficiently transform $k$ synchronization errors into $(1+\epsilon)k$ half-errors. This powerful new technique has many applications. In this paper, we focus on designing insdel codes, i.e., error correcting block codes (ECCs) for insertion deletion channels. While ECCs for both half-errors and synchronization errors have been intensely studied, the later has largely resisted progress. Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed by Schulman and Zuckerman. Insdel codes for asymptotically large or small noise rates were given in 2016 by Guruswami et al. but these codes are still polynomially far from the optimal rate-distance tradeoff. This makes the understanding of insdel codes up to this work equivalent to what was known for regular ECCs after Forney introduced concatenated codes in his doctoral thesis 50 years ago. A direct application of our synchronization strings based indexing method gives a simple black-box construction which transforms any ECC into an equally efficient insdel code with a slightly larger alphabet size. This instantly transfers much of the highly developed understanding for regular ECCs over large constant alphabets into the realm of insdel codes. Most notably, we obtain efficient insdel codes which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound for the complete noise spectrum

Nonlinear brain dynamics as macroscopic manifestation of underlying many-body field dynamics

Neural activity patterns related to behavior occur at many scales in time and space from the atomic and molecular to the whole brain. Here we explore the feasibility of interpreting neurophysiological data in the context of many-body physics by using tools that physicists have devised to analyze comparable hierarchies in other fields of science. We focus on a mesoscopic level that offers a multi-step pathway between the microscopic functions of neurons and the macroscopic functions of brain systems revealed by hemodynamic imaging. We use electroencephalographic (EEG) records collected from high-density electrode arrays fixed on the epidural surfaces of primary sensory and limbic areas in rabbits and cats trained to discriminate conditioned stimuli (CS) in the various modalities. High temporal resolution of EEG signals with the Hilbert transform gives evidence for diverse intermittent spatial patterns of amplitude (AM) and phase modulations (PM) of carrier waves that repeatedly re-synchronize in the beta and gamma ranges at near zero time lags over long distances. The dominant mechanism for neural interactions by axodendritic synaptic transmission should impose distance-dependent delays on the EEG oscillations owing to finite propagation velocities. It does not. EEGs instead show evidence for anomalous dispersion: the existence in neural populations of a low velocity range of information and energy transfers, and a high velocity range of the spread of phase transitions. This distinction labels the phenomenon but does not explain it. In this report we explore the analysis of these phenomena using concepts of energy dissipation, the maintenance by cortex of multiple ground states corresponding to AM patterns, and the exclusive selection by spontaneous breakdown of symmetry (SBS) of single states in sequences.Comment: 31 page

Lossless and near-lossless source coding for multiple access networks

A multiple access source code (MASC) is a source code designed for the following network configuration: a pair of correlated information sequences {X-i}(i=1)(infinity), and {Y-i}(i=1)(infinity) is drawn independent and identically distributed (i.i.d.) according to joint probability mass function (p.m.f.) p(x, y); the encoder for each source operates without knowledge of the other source; the decoder jointly decodes the encoded bit streams from both sources. The work of Slepian and Wolf describes all rates achievable by MASCs of infinite coding dimension (n --> infinity) and asymptotically negligible error probabilities (P-e((n)) --> 0). In this paper, we consider the properties of optimal instantaneous MASCs with finite coding dimension (n 0) performance. The interest in near-lossless codes is inspired by the discontinuity in the limiting rate region at P-e((n)) = 0 and the resulting performance benefits achievable by using near-lossless MASCs as entropy codes within lossy MASCs. Our central results include generalizations of Huffman and arithmetic codes to the MASC framework for arbitrary p(x, y), n, and P-e((n)) and polynomial-time design algorithms that approximate these optimal solutions

Quickest Sequence Phase Detection

A phase detection sequence is a length-$n$ cyclic sequence, such that the location of any length-$k$ contiguous subsequence can be determined from a noisy observation of that subsequence. In this paper, we derive bounds on the minimal possible $k$ in the limit of $n\to\infty$, and describe some sequence constructions. We further consider multiple phase detection sequences, where the location of any length-$k$ contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences. We study the optimal trade-offs between the lengths of the sequences, and describe some sequence constructions. We compare these phase detection problems to their natural channel coding counterparts, and show a strict separation between the fundamental limits in the multiple sequence case. Both adversarial and probabilistic noise models are addressed.Comment: To appear in the IEEE Transactions on Information Theor