A sequence of one-point codes from a tower of function fields

Asymptotics for the genus and the number of rational places in towers of function fields over a finite field

In this paper, we investigate a special tower of function fields, which is asymptotically bad. The primary goal is to construct explicitly a class of algebraic-geometry (AG) codes based on this tower. The so called Hermitian codes are codes corresponding to the 2nd function field in this tower.

In this paper we study general conditions to prove the infiniteness of the genus of certain towers of function fields over a perfect field. We show that many known examples of towers with infinite genus are particular cases of these conditions.

In this paper, a convenient set of functions is identified whose span includes all functions in a tower of function fields of Garcia and Stichtenoth that have poles only at the place at infinity. The latter set is of interest in the construction of long and efficient “one-point” AG codes.

Establishing Safety for Heavy Duty Vehicle Platooning: A Game Theoretical Approach

Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.

Image distortion analysis is a fundamental issue in many image processing problems, including compression, restoration, recognition, classification, and retrieval. Traditional image distortion evaluation approaches tend to be heuristic and are often limited to specific application environment. In this work, we investigate the problem of image distortion measurement based on the theory of Kolmogorov complexity, which has rarely been studied in the context of image processing. This work is motivated by the normalized information distance (NID) measure that has been shown to be a valid and universal distance metric applicable to similarity measurement of any two objects (Li et al. in IEEE Trans Inf Theory 50:3250–3264, 2004). Similar to Kolmogorov complexity, NID is non-computable. A useful practical solution is to approximate it using normalized compression distance (NCD) (Li et al. in IEEE Trans Inf Theory 50:3250–3264, 2004), which has led to impressive results in many applications such as construction of phylogeny trees using DNA sequences (Li et al. in IEEE Trans Inf Theory 50:3250–3264, 2004). In our earlier work, we showed that direct use of NCD on image processing problems is difficult and proposed a normalized conditional compression distance (NCCD) measure (Nikvand and Wang, 2010), which has significantly wider applicability than existing image similarity/distortion measures. To assess the distortions between two images, we first transform them into the wavelet transform domain. Assuming stationarity and good decorrelation of wavelet coefficients beyond local regions and across wavelet subbands, the Kolmogorov complexity may be approximated using Shannon entropy (Cover et al. in Elements of information theory. Wiley-Interscience, New York, 1991). Inspired by Sheikh and Bovik (IEEE Trans Image Process 15(2):430–444, 2006), we adopt a Gaussian scale mixture model for clusters of neighboring wavelet coefficients and a Gaussian channel model for the noise distortions in the human visual system. Combining these assumptions with the NID framework, we derive a novel normalized perceptual information distance measure, where maximal likelihood estimation and least square regression are employed for parameter fitting. We validate the proposed distortion measure using three large-scale, publicly available, and subject-rated image databases, which include a wide range of practical image distortion types and levels. Our results demonstrate the good prediction power of the proposed method for perceptual image distortions.

The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.

In this chapter, we establish conditions on the parameters of codes. In the first part of the chapter, we present bounds that relate between the length n , size M , minimum distance d , and the alphabet size q of a code. Two of these bounds—the Singleton bound and the sphere-packing bound—imply necessary conditions on the values of n, M, d , and q , so that a code with the respective parameters indeed exists. We also exhibit families of codes that attain each of these bounds. The third bound which we present—the Gilbert–Varshamov bound—is an existence result: it states that there exists a linear [ n, k, d ] code over GF( q ) whenever n, k, d , and q satisfy a certain inequality. Additional bounds are included in the problems at the end of this chapter. We end this part of the chapter by introducing another example of necessary conditions on codes—now in the form of MacWilliams' identities, which relate the distribution of the Hamming weights of the codewords in a linear code with the respective distribution in the dual code. The second part of this chapter deals with asymptotic bounds, which relate the rate of a code to its relative minimum distance δ = d/n and its alphabet size, as the code length n tends to infinity. In the third part of the chapter, we shift from the combinatorial setting of ( n, M, d ) codes to the probabilistic framework of the memoryless q -ary symmetric channel.

The separation system is crucial for the launch of satellites. Dynamic characteristics of satellite separation are quite complex and difficult to predict. With respect to the helical compression spring mechanism, an approach using transient perturbation analysis is presented. The separation springs and limit switches are mathematically modeled, and disturbing forces and moments are considered. ADAMS and MATLAB software platforms are combined to obtain separation trajectory and attitude parameters. The minimum relative distance is proposed to show whether there is collision between the satellite and launch vehicles. Emphasis is placed on introducing the approach by analyzing a typical separation system. With experimental design and statistical analysis, the influences of perturbation factors are concluded. For example, three angular velocities are approximately linear with center of gravity offsets of the satellite and deviations of spring parameters; however, the effect law of asynchronous time is non-linear. A ground test system of satellite separation is designed and the test results are compared with the analysis, which prove accuracy of the dynamic model and feasibility of the approach.

The multi-objective optimization of linearized impulsive rendezvous is investigated in this paper and this optimization includes the minimum characteristic velocity, the minimum time of flight, and the maximum safety performance index, of which the trajectory safety performance index is defined as the minimum relative distance between a chaser and a target in the chaser's free-flying path. A theoretical model for calculating this safety performance index is provided. The three-objective optimization model is proposed based on the Clohessy-Wiltshire system, wherein a generalized inverse matrix solution for linear equations is applied to avoid handling the terminal equality constraints. The multi-objective nondominated sorting genetic algorithm is employed to obtain the Pareto solution set. The proposed approach is evaluated using the -V-bar homing and +V-bar homing rendezvous missions. It is shown that tradeoffs between time of flight, fuel cost, and passive trajectory safety for rendezvous trajectory is quickly demonstrated by the approach. By identifying multiple solutions, the approach can produce a variety of missions to meet different needs.

The minimum distance of some families of expander codes is studied, as well as some related families of codes defined on bipartite graphs. The weight spectrum and the minimum distance of a random ensemble of such codes are computed and it is shown that it sometimes meets the Gilbert-Varshamov (GV) bound. A lower bound on the minimum distances of constructive families of expander codes is derived. The relative minimum distance of the expander code is shown to exceed the product bound, i.e., the quantity /spl delta//sub 0//spl delta//sub 1/ where /spl delta//sub 0/ and /spl delta//sub 1/ are the minimum relative distances of the constituent codes. As a consequence of this, a polynomially constructible family of expander codes is obtained whose relative distance exceeds the Zyablov bound on the distance of serial concatenations.

We consider two-dimensional states of matter satisfying an uniform area law\nfor entanglement. We show that the topological entanglement entropy is equal to\nthe minimum relative entropy distance from the reduced state to the set of\nthermal states of local models. The argument is based on strong subadditivity\nof quantum entropy. For states with zero topological entanglement entropy, in\nparticular, the formula gives locality of the states at the boundary of a\nregion as thermal states of local Hamiltonians. It also implies that the\nentanglement spectrum of a two-dimensional region is equal to the spectrum of a\none-dimensional local thermal state on the boundary of the region.\n

Minimal codewords were introduced by Massey (Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, pp 276---279, 1993) for cryptographical purposes. They are used in particular secret sharing schemes, to model the access structures. We study minimal codewords of weight smaller than 3 · 2 m?r in binary Reed---Muller codes RM(r, m) and translate our problem into a geometrical one, using a classification result of Kasami and Tokura (IEEE Trans Inf Theory 16:752---759, 1970) and Kasami et al. (Inf Control 30(4):380---395, 1976) on Boolean functions. In this geometrical setting, we calculate numbers of non-minimal codewords. So we obtain the number of minimal codewords in the cases where we have information about the weight distribution of the code RM(r, m). The presented results improve previous results obtained theoretically by Borissov et al. (Discrete Appl Math 128(1), 65---74, 2003), and computer aided results of Borissov and Manev (Serdica Math J 30(2-3), 303---324, 2004). This paper is in fact an extended abstract. Full proofs can be found on the arXiv.

In this paper we study the problem of constructing non-trivial subtowers and supertowers of recursive towers of function fields over finite fields.