Perfect Mannheim, Lipschitz and Hurwitz weight codes

In this paper, upper bounds on codes over Gaussian integers, Lipschitz integers and Hurwitz integers with respect to Mannheim metric, Lipschitz and Hurwitz metric are given.Comment: 21 page

Perfect 1-error-correcting Lipschitz weight codes

Let $pi$ be a Lipschitz prime and $p=pipi^star$. Perfect 1-error-correcting codes in $H(mathbb{Z})_pi^n$ are constructed for every prime number $pequiv1(bmod;4)$. This completes a result of the authors in an earlier work, emph{Perfect Mannheim, Lipschitz and Hurwitz weight codes}, (Mathematical Communications, Vol 19, No 2, pp. 253 -- 276 (2014)), where a construction is given in the case $pequiv3,(bmod;4)$

Perfect 1-error-correcting Hurwitz weight codes

Let( $pi$) be a Hurwitz prime and ($p = pi pi ^star$). In this paper, we construct perfect 1-error-correcting codes in ($cal{H}_{pi}^n$) for every prime number ($p > 3$), where ($cal{H}$) denotes the set of Hurwitz integers

Perfect 1-error-correcting Hurwitz weight codes

Let( $pi$) be a Hurwitz prime and ($p = pi pi ^star$). In this paper, we construct perfect 1-error-correcting codes in ($cal{H}_{pi}^n$) for every prime number ($p > 3$), where ($cal{H}$) denotes the set of Hurwitz integers

Codes over Hurwitz integers

In this study, we obtain new classes of linear codes over Hurwitz integers equipped with a new metric. We refer to the metric as Hurwitz metric. The codes with respect to Hurwitz metric use in coded modu- lation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither Hamming metric nor Lee metric. Also, we define decoding algorithms for these codes when up to two coordinates of a transmitted code vector are effected by error of arbitrary Hurwitz weight.Comment: 11 page

Online Learning of Quantum States

Suppose we have many copies of an unknown $n$-qubit state $\rho$. We measure some copies of $\rho$ using a known two-outcome measurement $E_{1}$, then other copies using a measurement $E_{2}$, and so on. At each stage $t$, we generate a current hypothesis $\sigma_{t}$ about the state $\rho$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\operatorname{Tr}(E_{i} \sigma_{t}) - \operatorname{Tr}(E_{i}\rho) |$, the error in our prediction for the next measurement, is at least $\varepsilon$ at most $\operatorname{O}\!\left(n / \varepsilon^2 \right)$ times. Even in the "non-realizable" setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that do significantly worse than the best possible states at most $\operatorname{O}\!\left(\sqrt {Tn}\right)$ times on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.Comment: 18 page