Relative Minimum Distance Research Articles

Asymptotic bound on binary self-orthogonal codes

We present two constructions for binary self-orthogonal codes. It turns out that our constructions yield a constructive bound on binary self-orthogonal codes. In particular, when the information rate R = 1/2, by our constructive lower bound, the relative minimum distance δ ≈ 0.0595 (for GV bound, δ ≈ 0.110). Moreover, we have proved that the binary self-orthogonal codes asymptotically achieve the Gilbert-Varshamov bound.

Improved Asymptotic Bounds for Codes Using Distinguished Divisors of Global Function Fields

For a prime power q, let $\alpha_q$ be the standard function in the asymptotic theory of codes, that is, $\alpha_q(\delta)$ is the largest asymptotic information rate that can be achieved for a given asymptotic relative minimum distance $\delta$ of q-ary codes. In recent years the Tsfasman–Vlăduţ–Zink lower bound on $\alpha_q(\delta)$ was improved by Elkies, Xing, Niederreiter and Özbudak, and Maharaj. In this paper we show further improvements on these bounds by using distinguished divisors of global function fields. We also show improved lower bounds on the corresponding function $\alpha_q^{\rm lin}$ for linear codes.

Robustness of Quantum Markov Chains

If the conditional information of a classical probability distribution of three random variables is zero, then it obeys a Markov chain condition. If the conditional information is close to zero, then it is known that the distance (minimum relative entropy) of the distribution to the nearest Markov chain distribution is precisely the conditional information. We prove here that this simple situation does not obtain for quantum conditional information. We show that for tri-partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum Markov chain state, but the distance can be much greater; indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor.

Analytic formulations for calculating nearly singular integrals in two-dimensional BEM

There exist the nearly singular integrals in the boundary integral equations when a source point is close to an integration element but not on the element, such as the field problems with thin domains. In this paper, the analytic formulations are achieved to calculate the nearly weakly singular, strongly singular and hyper-singular integrals on the straight elements for the two-dimensional (2D) boundary element methods (BEM). The algorithm is performed after the BIE are discretized by a set of boundary elements. The singular factor, which is expressed by the minimum relative distance from the source point to the closer element, is separated from the nearly singular integrands by the use of integration by parts. Thus, it results in exact integrations of the nearly singular integrals for the straight elements, instead of the numerical integration. The analytic algorithm is also used to calculate nearly singular integrals on the curved element by subdividing it into several linear or sub-parametric elements only when the nearly singular integrals need to be determined. The approach can achieve high accuracy in cases of the curved elements without increasing other computational efforts. As an application, the technique is employed to analyze the 2D elasticity problems, including the thin-walled structures. Some numerical results demonstrate the accuracy and effectiveness of the algorithm.

Optimal Multi-Objective Nonlinear Impulsive Rendezvous

The multi-objective optimization of unperturbed two-body impulsive rendezvous is investigated in this paper. In addition to the total characteristic velocity and the time of rendezvous flight, the minimum relative distance between the chaser and the target in the chaser's free-flying path is proposed as another design performance index. The feasible iteration optimization model of three-objective impulsive rendezvous with path constraints including trajectory safety and field of view constraints is established using a Lambert algorithm. The multi-objective nondominated sorting genetic algorithm is employed to obtain the Pareto solution set. The -V-bar and +V-bar homing rendezvous missions are examined by the proposed approach. It is shown that tradeoffs between time of flight, propellant cost and passive trajectory safety, and several inherent principles of the rendezvous trajectory are demonstrated by this approach.

Optimal Multi-Objective Linearized Impulsive Rendezvous

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.

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

We construct a sequence of one-point codes from a tower of function fields whose relative minimum distances have a positive limit. Our tower is characterized by principal divisors. We determine completely the minimum distance of the codes from the first field of our tower. These results extend those of Stichtenoth [IEEE Trans Inform Theory (1988), 34(15):1345---1348], Yang and Kumar [Lecture Notes in Mathematics, 1518, (1991), Springer-Verlag, Berlin Hidelberg New York, pp. 99---107], and Garcia [Comm. Algebra, 20(12): 3683---3689]. As an application, we show that the minimum distance corresponds to the Feng---Rao bound.

Improved Nearly-MDS Expander Codes

A construction of expander codes is presented with the following three properties: (i) the codes lie close to the Singleton bound, (ii) they can be encoded in time complexity that is linear in their code length, and (iii) they have a linear-time bounded-distance decoder. By using a version of the decoder that corrects also erasures, the codes can replace MDS outer codes in concatenated constructions, thus resulting in linear-time encodable and decodable codes that approach the Zyablov bound or the capacity of memoryless channels. The presented construction improves on an earlier result by Guruswami and Indyk in that any rate and relative minimum distance that lies below the Singleton bound is attainable for a significantly smaller alphabet size.

Upper bounds on the rate of LDPC codes as a function of minimum distance

New upper bounds on the rate of low-density parity-check (LDPC) codes as a function of the minimum distance of the code are derived. The bounds apply to regular LDPC codes, and sometimes also to right-regular LDPC codes. Their derivation is based on combinatorial arguments and linear programming. The new bounds improve upon the previous bounds due to Burshtein et al. It is proved that at least for high rates, regular LDPC codes with full-rank parity-check matrices have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound.

Distance properties of expander codes

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.

Further improvements on asymptotic bounds for codes using distinguished divisors

For a prime power q, let α q be the standard function in the asymptotic theory of codes, that is, α q ( δ ) is the largest asymptotic information rate that can be achieved for a given asymptotic relative minimum distance δ of q-ary codes. In recent years the Tsfasman–Vlăduţ–Zink lower bound on α q ( δ ) was improved by Elkies, Xing, and Niederreiter and Özbudak. In this paper we show further improvements on these bounds by using distinguished divisors of global function fields.

On the relative distances of six points in a plane convex body

Let C be a convex body in the Euclidean plane. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body.

Endcoding Complexity Versus Minimum Distance

A bound on the minimum distance of a binary error-correcting code is established given constraints on the computational time-space complexity of its encoder where the encoder is modeled as a branching program. The bound obtained asserts that if the encoder uses linear time and sublinear memory in the most general sense, then the minimum distance of the code cannot grow linearly with the block length when the rate is nonvanishing, that is, the minimum relative distance of the code tends to zero in such a setting. The setting is general enough to include nonserially concatenated turbo-like codes and various generalizations. Our argument is based on branching program techniques introduced by Ajtai. The case of constant-depth AND-OR circuit encoders with unbounded fanins are also considered.

Interactions of boundary shear stress, secondary currents and velocity

This paper deals with the interaction of boundary shear stress, velocity distribution and secondary currents in open channel flows. A method for computing boundary shear stress and velocity distribution in steady, uniform and fully developed turbulent flows is developed by applying an order-of-magnitude analysis to the Reynolds equations. A simplified relationship between the wall-tangential and wall-normal terms in the Reynolds equation is hypothesized, then the Reynolds equations become solvable. This analysis suggests that the energy from the main flow is transported towards the nearest boundary to be dissipated through a minimum relative distance or normal distance of the boundary. The equations governing the boundary shear stress and Reynolds shear stress distributions are obtained, and the influence of wall-normal velocity on the streamwise velocity is assessed. It is found that the classical log-law is valid only when the wall-normal velocity is zero, the non-zero wall-normal velocity results in the derivation of measured streamwise velocity from the classical log-law. The derived equations are in good agreement with existing experimental data available in the literature.

Estimation of diffusion coefficients, lateral shear stress, and velocity in open channels with complex geometry

The lateral distributions of depth‐averaged apparent shear stress, depth mean velocity, and diffusion coefficients are essential in certain quantitative analysis for sediment transport and environmental studies. An analytical method for the computation of these parameters is presented. A mathematical relationship between these parameters, based on the concept of surplus energy transport through a minimum relative distance developed by Yang and Lim [1997], is established, the depth‐averaged apparent shear stress is determined from the boundary shear stress, depth mean velocity is obtained by considering the influence of nonuniform shear velocity and the free surface in 3‐D channels, and the diffusion coefficients are linked to the depth‐averaged apparent shear and velocity. The theoretical formulations for the distributions of depth‐averaged apparent shear stresses, depth mean velocity and diffusion coefficients in trapezoidal and compound channels are presented. Comparisons between the theoretical and the measured lateral distributions of the depth‐averaged apparent shear stresses and the depth mean velocities are also presented, and a reasonable agreement is achieved.

Source and channel rate allocation for channel codes satisfying the Gilbert-Varshamov or Tsfasman-Vladut-Zink bounds

We derive bounds for optimal rate allocation between source and channel coding for linear channel codes that meet the Gilbert-Varshamov or Tsfasman-Vladut-Zink (1984) bounds. Formulas giving the high resolution vector quantizer distortion of these systems are also derived. In addition, we give bounds on how far below the channel capacity the transmission rate should be for a given delay constraint. The bounds obtained depend on the relationship between channel code rate and relative minimum distance guaranteed by the Gilbert-Varshamov bound, and do not require sophisticated decoding beyond the error correction limit. We demonstrate that the end-to-end mean-squared error decays exponentially fast as a function of the overall transmission rate, which need not be the case for certain well-known structured codes such as Hamming codes.

BOUNDARY SHEAR STRESS DISTRIBUTIONS IN SMOOTH RETANGULAR OPEN CHANNEL FLOWS.

Keywords NANYANG TECHNOLOGICAL UNIVERSITY, SINGAPORE SHEAR, STRESS, DISTRIBUTION, BOUNDARIES, SMOOTH, RECTANGULAR, OPEN, CHANNELS, FLOW, METHODS, ANALYTICAL, FLUIDS, THREE DIMENSIONAL, PERIMETER, WETTED, STEADY, UNIFORM, ASPECT, RATIOS, MODELS, MATHEMATICAL, HYDRAULICS, HYDRODYNAMICS, RIVERS, THEORY, MECHANISMS, ENERGY, TRANSPORTATION, INFLUENCE, ROUGHNESS, DISSIPATION... Show All

Mechanism of Energy Transportation and Turbulent Flow in a 3D Channel

This paper presents an analytical approach for the partitioning of the flow cross-sectional area in steady and uniform three-dimensional channels. The mechanism and direction of energy transport from the main flow are analyzed first. Based on the availability of surplus energy from the main flow, a concept of energy transportation through a minimum relative distance toward a unit area on the wetted perimeter is proposed. This leads to a novel method to analytically divide the flow area into various parts corresponding to the channel shape and roughness composition of its wetted perimeter. The demarcated boundary or “division line” can be evaluated using a presented equation. The existence of the division lines, which are lines of zero Reynolds shear stress within the flow region, has been proven through comparison with turbulence characteristics measurements. The method is illustrated in a study of the shear stress distribution in smooth rectangular open channels. Analytical solutions, valid for all aspect ratios, are derived for the mean side wall and bed shear stresses, and they compared well with existing experimental data from various researchers.

Improved Gilbert-Varshamov bound for constrained systems

Nonconstructive existence results are obtained for block error-correcting codes whose codewords lie in a given constrained system. Each such system is defined as a set of words obtained by reading the labels of a finite directed labeled graph. For a prescribed constrained system and relative minimum distance delta , the new lower bounds on the rate of such codes improve on those derived recently by V.D. Kolesnik and V.Y. Krachkovsky (1991). The better bounds are achieved by considering a special subclass of sequences in the constrained system, namely, those having certain empirical statistics determined by delta .< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On repeated-root cyclic codes

A parity-check matrix for a q-ary repeated-root cyclic code is derived using the Hasse derivative. Then the minimum distance of a q-ary repeated-root cyclic code is expressed in terms of the minimum distance of a certain simple-root cyclic code. With the help of this result, several binary repeated-root cyclic codes of lengths up to n=62 are shown to contain the largest known number of codewords for their given length and minimum distance. The relative minimum distance d/sub min//n of q-ary repeated-root cyclic codes of rate r>or=R is proven to tend to zero as the largest multiplicity of a root of the generator g(x) increases to infinity. It is further shown that repeated-root cycle codes cannot be asymptotically better than simple-root cyclic codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>