Quantum generalized Reed-Solomon codes: Unified framework for quantum MDS codes

We construct a new family of quantum MDS codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them analytically. We find that existing quantum MDS codes can be unified under these codes in the sense that when a quantum MDS code exists, then a quantum code of this type with the same parameters also exists. Thus as far as is known at present, they are the most important family of quantum MDS codes.Comment: 9 pages, no figure

Contextuality in Measurement-based Quantum Computation

We show, under natural assumptions for qubit systems, that measurement-based quantum computations (MBQCs) which compute a non-linear Boolean function with high probability are contextual. The class of contextual MBQCs includes an example which is of practical interest and has a super-polynomial speedup over the best known classical algorithm, namely the quantum algorithm that solves the Discrete Log problem.Comment: Version 3: probabilistic version of Theorem 1 adde

Flow convergence routing hypothesis for pool-riffle maintenance in alluvial rivers

The velocity reversal hypothesis is commonly cited as a mechanism for the maintenance of pool-riffle morphology. Although this hypothesis is based on the magnitude of mean flow parameters, recent studies have suggested that mean parameters are not sufficient to explain the dominant processes in many pool-riffle sequences. In this study, two- and three-dimensional models are applied to simulate flow in the pool-riffle sequence on Dry Creek, California, where the velocity reversal hypothesis was first proposed. These simulations provide an opportunity to evaluate the hydrodynamics underlying the observed reversals in near-bed and section-averaged velocity and are used to investigate the influence of secondary currents, the advection of momentum, and cross-stream flow variability. The simulation results support the occurrence of a reversal in mean velocity and mean shear stress with increasing discharge. However, the results indicate that the effects of flow convergence due to an upstream constriction and the routing of flow through the system are more significant in influencing pool-riffle morphology than the occurrence of a mean velocity reversal. The hypothesis of flow convergence routing is introduced as a more meaningful explanation of the mechanisms acting to maintain pool-riffle morphology

Complete Solving for Explicit Evaluation of Gauss Sums in the Index 2 Case

Let $p$ be a prime number, $q=p^f$ for some positive integer $f$, $N$ be a positive integer such that $\gcd(N,p)=1$, and let \k be a primitive multiplicative character of order $N$ over finite field \fq. This paper studies the problem of explicit evaluation of Gauss sums in "\textsl{index 2 case}" (i.e. f=\f{\p(N)}{2}=[\zn:\pp], where \p(\cd) is Euler function). Firstly, the classification of the Gauss sums in index 2 case is presented. Then, the explicit evaluation of Gauss sums G(\k^\la) (1\laN-1) in index 2 case with order $N$ being general even integer (i.e. N=2^{r}\cd N_0 where $r,N_0$ are positive integers and $N_03$ is odd.) is obtained. Thus, the problem of explicit evaluation of Gauss sums in index 2 case is completely solved

From Skew-Cyclic Codes to Asymmetric Quantum Codes

We introduce an additive but not $\mathbb{F}_4$-linear map $S$ from $\mathbb{F}_4^{n}$ to $\mathbb{F}_4^{2n}$ and exhibit some of its interesting structural properties. If $C$ is a linear $[n,k,d]_4$-code, then $S(C)$ is an additive $(2n,2^{2k},2d)_4$-code. If $C$ is an additive cyclic code then $S(C)$ is an additive quasi-cyclic code of index $2$. Moreover, if $C$ is a module $\theta$-cyclic code, a recently introduced type of code which will be explained below, then $S(C)$ is equivalent to an additive cyclic code if $n$ is odd and to an additive quasi-cyclic code of index $2$ if $n$ is even. Given any $(n,M,d)_4$-code $C$, the code $S(C)$ is self-orthogonal under the trace Hermitian inner product. Since the mapping $S$ preserves nestedness, it can be used as a tool in constructing additive asymmetric quantum codes.Comment: 16 pages, 3 tables, submitted to Advances in Mathematics of Communication

Quantitative analysis of cell types during growth and morphogenesis in Hydra

Tissue maceration was used to determine the absolute number and the distribution of cell types in Hydra. It was shown that the total number of cells per animal as well as the distribution of cells vary depending on temperature, feeding conditions, and state of growth. During head and foot regeneration and during budding the first detectable change in the cell distribution is an increase in the number of nerve cells at the site of morphogenesis. These results and the finding that nerve cells are most concentrated in the head region, diminishing in density down the body column, are discussed in relation to tissue polarity

On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension $n$ consisting of $n^2$ operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in \C^n which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.Comment: 29 pages, LaTe

The Glasgow-Maastricht foot model, evaluation of a 26 segment kinematic model of the foot

BACKGROUND: Accurately measuring of intrinsic foot kinematics using skin mounted markers is difficult, limited in part by the physical dimensions of the foot. Existing kinematic foot models solve this problem by combining multiple bones into idealized rigid segments. This study presents a novel foot model that allows the motion of the 26 bones to be individually estimated via a combination of partial joint constraints and coupling the motion of separate joints using kinematic rhythms. METHODS: Segmented CT data from one healthy subject was used to create a template Glasgow-Maastricht foot model (GM-model). Following this, the template was scaled to produce subject-specific models for five additional healthy participants using a surface scan of the foot and ankle. Forty-three skin mounted markers, mainly positioned around the foot and ankle, were used to capture the stance phase of the right foot of the six healthy participants during walking. The GM-model was then applied to calculate the intrinsic foot kinematics. RESULTS: Distinct motion patterns where found for all joints. The variability in outcome depended on the location of the joint, with reasonable results for sagittal plane motions and poor results for transverse plane motions. CONCLUSIONS: The results of the GM-model were comparable with existing literature, including bone pin studies, with respect to the range of motion, motion pattern and timing of the motion in the studied joints. This novel model is the most complete kinematic model to date. Further evaluation of the model is warranted

Information-bit error rate and false positives in an MDS code

In this paper, a refinement of the weight distribution in an MDS code is computed. Concretely, the number of codewords with a fixed amount of nonzero bits in both information and redundancy parts is obtained. This refinement improves the theoretical approximation of the information-bit and -symbol error rate, in terms of the channel bit-error rate, in a block transmission through a discrete memoryless channel. Since a bounded distance reproducing encoder is assumed, the computation of the here-called false positive (a decoding failure with no information-symbol error) is provided. As a consequence, a new performance analysis of an MDS code is proposed