Computing the Ball Size of Frequency Permutations under Chebyshev Distance

Let $S_n^\lambda$ be the set of all permutations over the multiset $\{\overbrace{1,...,1}^{\lambda},...,\overbrace{m,...,m}^\lambda\}$ where $n=m\lambda$. A frequency permutation array (FPA) of minimum distance $d$ is a subset of $S_n^\lambda$ in which every two elements have distance at least $d$. FPAs have many applications related to error correcting codes. In coding theory, the Gilbert-Varshamov bound and the sphere-packing bound are derived from the size of balls of certain radii. We propose two efficient algorithms that compute the ball size of frequency permutations under Chebyshev distance. Both methods extend previous known results. The first one runs in $O({2d\lambda \choose d\lambda}^{2.376}\log n)$ time and $O({2d\lambda \choose d\lambda}^{2})$ space. The second one runs in $O({2d\lambda \choose d\lambda}{d\lambda+\lambda\choose \lambda}\frac{n}{\lambda})$ time and $O({2d\lambda \choose d\lambda})$ space. For small constants $\lambda$ and $d$, both are efficient in time and use constant storage space.Comment: Submitted to ISIT 201

LP-decodable multipermutation codes

In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are permutations of a multiset that may consist of duplicate entries. We first introduce a new class of matrices called multipermutation matrices. We characterize the convex hull of multipermutation matrices. Based on this characterization, we propose a new class of codes that we term LP-decodable multipermutation codes. Then, we derive two LP decoding algorithms. We first formulate an LP decoding problem for memoryless channels. We then derive an LP algorithm that minimizes the Chebyshev distance. Finally, we show a numerical example of our algorithm.Comment: This work was supported by NSF and NSERC. To appear at the 2014 Allerton Conferenc

Machine Learning Assisted Many-Body Entanglement Measurement

Entanglement not only plays a crucial role in quantum technologies, but is key to our understanding of quantum correlations in many-body systems. However, in an experiment, the only way of measuring entanglement in a generic mixed state is through reconstructive quantum tomography, requiring an exponential number of measurements in the system size. Here, we propose a machine learning assisted scheme to measure the entanglement between arbitrary subsystems of size $N_A$ and $N_B$, with $\mathcal{O}(N_A + N_B)$ measurements, and without any prior knowledge of the state. The method exploits a neural network to learn the unknown, non-linear function relating certain measurable moments and the logarithmic negativity. Our procedure will allow entanglement measurements in a wide variety of systems, including strongly interacting many body systems in both equilibrium and non-equilibrium regimes.Comment: 16 pages, 10 figures, including appendi

Constructions of Rank Modulation Codes

Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with the Kendall tau distance. We present several general constructions of codes in permutations that cover a broad range of code parameters. In particular, we show a number of ways in which conventional error-correcting codes can be modified to correct errors in the Kendall space. Codes that we construct afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the Hamming metric. For instance, from binary BCH codes we obtain codes correcting $t$ Kendall errors in $n$ memory cells that support the order of $n!/(\log_2n!)^t$ messages, for any constant $t= 1,2,...$ We also construct families of codes that correct a number of errors that grows with $n$ at varying rates, from $\Theta(n)$ to $\Theta(n^{2})$. One of our constructions gives rise to a family of rank modulation codes for which the trade-off between the number of messages and the number of correctable Kendall errors approaches the optimal scaling rate. Finally, we list a number of possibilities for constructing codes of finite length, and give examples of rank modulation codes with specific parameters.Comment: Submitted to IEEE Transactions on Information Theor

Lossy compression of permutations

We investigate the lossy compression of permutations by analyzing the trade-off between the size of a source code and the distortion with respect to Kendall tau distance, Spearman's footrule, Chebyshev distance and ℓ[subscript 1] distance of inversion vectors. We show that given two permutations, Kendall tau distance upper bounds the ℓ[subscript 1] distance of inversion vectors and a scaled version of Kendall tau distance lower bounds the ℓ[subscript 1] distance of inversion vectors with high probability, which indicates an equivalence of the source code designs under these two distortion measures. Similar equivalence is established for all the above distortion measures, every one of which has different operational significance and applications in ranking and sorting. These findings show that an optimal coding scheme for one distortion measure is effectively optimal for other distortion measures above.United States. Air Force Office of Scientific Research (Grant FA9550-11-1-0183)National Science Foundation (U.S.) (Grant CCF-1017772