A note on the construction of optimal linear codes

Let us denote by (n,k,d)-code, a binary linear code with code length nk information symbols and the minimum distance d. It is well known that the problem of obtaining a binary linear code whose code length n is minimum among (n,k,d)-codes for given integers k and d, is equivalent to solve a linear programming whose solutions correspond to a minimum redundancy error-correcting code. In this paper it will be shown that for some given integers d, there exists no solution of the linear programming except a solution which is obtained using a flat in a finite projective geometry.     

Steiner triple systems of order 15 and their codes

The binary linear codes generated by incidence matrices of the 80 Steiner triple systems on 15 points (STS(15)) are studied. The 80 codes of length 35 spanned by incidence vectors of the points are all non-isomorphic. In contrast, a pair of codes of length 15 generated by blocks are isomorphic if and only if the corresponding incidence matrices have the same rank over GF(2). The weight distribution, the automorphism groups of the codes, and the distribution of the Steiner triple systems within the codes are computed. There are 54 codes of length 35 that contain several non-isomorphic STS(15)'s, and any such code is generated by an STS(15) of largest 2-rank.     

An application of difference sets to a problem concerning graphical codes

By using difference sets, we give an answer to the following problem concerning graphical codes: When is the binary code generated by the complete graph K_n contained in some binary Hamming code? It turns out that this holds if and only if n is one of the numbers 2, 3 and 6.     

A relation between the average Hamming distance and the average Hamming weight of binary codes

An upper bound is presented for the average Hamming distance (avd) of binary codes by using the average Hamming weight (avw) of the codes. The upper bound is attained by a large class of codes. Hence, for this class of codes, a relation between the avd and the avw of a code is established. The results are applied to the avd between two codes, similar results are obtained.     

Extendable Steiner designs from finite geometries

We describe a simple procedure for constructing Steiner 2-designs with the parameters of the designs of points and lines of a finite projective or affine geometry of dimension m⩾3; the codes of many of the designs constructed in this way will contain the code of the relevant finite-geometry design (a Reed-Muller or generalized Reed-Muller code). The designs can be extended to 3-designs provided that planes in the finite-geometry design extend.     

On the equivalence of certain constant weight codes and combinatorial designs

Stinson and van Rees (Combinatorica (1984), 357–362) proved that the existence of an equidistant code E: (n = 4s + 1,d = 2s, N) with N = n implies the existence of a certain symmetrical BIB design. This result is extended here for the constant weight codes with the same n and d. A theorem on the equivalence of a Hadamard matrix and a certain constant weight code is also proved.     

Batch sequential designs for computer experiments

Computer models simulating a physical process are used in many areas of science. Due to the complex nature of these codes it is often necessary to approximate the code, which is typically done using a Gaussian process. In many situations the number of code runs available to build the Gaussian process approximation is limited. When the initial design is small or the underlying response surface is complicated this can lead to poor approximations of the code output. In order to improve the fit of the model, sequential design strategies must be employed. In this paper we introduce two simple distance based metrics that can be used to augment an initial design in a batch sequential manner. In addition we propose a sequential updating strategy to an orthogonal array based Latin hypercube sample. We show via various real and simulated examples that the distance metrics and the extension of the orthogonal array based Latin hypercubes work well in practice.     

Existence of certain class of duadic group algebra codes

Duadic codes are defined in terms of idempotents of a group algebra GF(q)G, where G is a finite group and gcd(q,|G|)=1. Under the conditions of (1) q=2^m, and (2) the idempotents are taken to be central and (3) the splitting is μ₋₁, we show that such duadic codes exist if and only if q has odd-order modulo |G|.     

Branch and bound algorithms for maximizing expected improvement functions

Deterministic computer simulations are often used as replacement for complex physical experiments. Although less expensive than physical experimentation, computer codes can still be time-consuming to run. An effective strategy for exploring the response surface of the deterministic simulator is the use of an approximation to the computer code, such as a Gaussian process (GP) model, coupled with a sequential sampling strategy for choosing design points that can be used to build the GP model. The ultimate goal of such studies is often the estimation of specific features of interest of the simulator output, such as the maximum, minimum, or a level set (contour). Before approximating such features with the GP model, sufficient runs of the computer simulator must be completed.Sequential designs with an expected improvement (EI) design criterion can yield good estimates of the features with minimal number of runs. The challenge is that the expected improvement function itself is often multimodal and difficult to maximize. We develop branch and bound algorithms for efficiently maximizing the EI function in specific problems, including the simultaneous estimation of a global maximum and minimum, and in the estimation of a contour. These branch and bound algorithms outperform other optimization strategies such as genetic algorithms, and can lead to significantly more accurate estimation of the features of interest.     

Self-dual codes which are principal ideals of the group algebra F2[{F2m, + }]

This paper follows up Camion's contribution on self-dual codes which are principal ideals of the algebra F₂[{F_2^m, + }], the so-called H-codes. Our main result is that this class of codes does not meet the Gilbert-Varshamov bound. We obtain this result by giving an upper bound on the minimal distance of any H-code. We characterize extremal H-codes and link up their generators with certain difference sets.     

The minimum L2 distance estimator for Poisson mixture models

A robust estimator is developed for Poisson mixture models with a known number of components. The proposed estimator minimizes the L₂ distance between a sample of data and the model. When the component distributions are completely known, the estimators for the mixing proportions are in closed form. When the parameters for the component Poisson distributions are unknown, numerical methods are needed to calculate the estimators. Compared to the minimum Hellinger distance estimator, the minimum L₂ estimator can be less robust to extreme outliers, and often more robust to moderate outliers.     

Weight hierarchies of linear codes of dimension 3

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d₁,d₂,…,d_k), where d_r is the smallest support of an r-dimensional subcode of C. The weight hierarchies of [n,3;q] codes are studied. In particular, for q⩽5 the possible weight hierarchies of [n,3;q] codes are determined.     

Orthogonal diagonal Latin squares with small subsquares

We prove that there exists a pair of orthogonal diagonal Latin squares of order v with missing subsquares of side n(ODLS(v, n)) if and only if v ⩾ 3n + 2 and v − n even, in the cases n ⩽ 6.     

A property of coefficients of variation for ordered bivariate log-normal variables

This paper considers the maximum and minimum of a pair of log-normal variables with equal mean. It shows that either order statistic has a smaller coefficient of variation than the two original log-normal variables provided the latter are of equal variance. When the variances are unequal, as the variance ratio increases, the minimum (maximum), has a smaller coefficient of variation if the correlation coefficient of the log-normal variables is small (small) and the variances are large (small).     

Methodology and analysis for comparing discrete linear l1 approximation codes

This is the first of a projected series of papers dealing with computational experimentation in mathematical programming. This paper provides early results of a test case using four discrete linear L₁ approximation codes. Variables influencing code behavior are identified and measures of performance are specified. More importantly, an experimental design is developed for assessing code performance and is illustrated using the variable “problem size”.     

A technique for generating Gray codes

It is often desirable to use Gray codes with properties different from those of the standard binary reflected code. Previously, only short codes (32- to 256-element) could be systematically generated or found with a given set of properties. This paper describes a technique where long codes (65000-element or longer) as well as short codes can systematically be generated with desired properties. The technique is described and demostrated by generating codes of various lenght with the desired property of equal column change counts. Several examples of the use of the technique for generating codes with other desired properties are outlined.     

A fast robust method for fitting gamma distributions

The art of fitting gamma distributions robustly is described. In particular we compare methods of fitting via minimizing a Cramér Von Mises distance, an L ₂ minimum distance estimator, and fitting a B-optimal M-estimator. After a brief prelude on robust estimation explaining the merits in terms of weak continuity and Fréchet differentiability of all the aforesaid estimators from an asymptotic point of view, a comparison is drawn with classical estimation and fitting. In summary, we give a practical example where minimizing a Cramér Von Mises distance is both efficacious in terms of efficiency and robustness as well as being easily implemented. Here gamma distributions arise naturally for “in control” representation indicators from measurements of spectra when using fourier transform infrared (FTIR) spectroscopy. However, estimating the in-control parameters for these distributions is often difficult, due to the occasional occurrence of outliers.     

Asymptotic normality of the minimum distance estimators for a Poisson process with a discontinuous intensity function

We consider the problem of estimation of a two-dimensional parameter θ₀=(θ₁,θ₂) of a Poisson process. The intensity function of the process is a smooth function with respect to θ₁ and is a discontinuous function of θ₂. We show the consistency and asymptotic normality of the minimum distance estimator of θ₀.     

Finding the best treatment under heavy censoring and hidden bias

Summary.  We analyse male survival duration after hospitalization following an acute myocardial infarction with a large ( N =11024) Finnish data set to find the best performing hospital district (and to disseminate its treatment protocol). This is a multiple-treatment problem with 21 treatments (i.e. 21 hospital districts). The task of choosing the best treatment is difficult owing to heavy right censoring (73%), which makes the usual location measures (the mean and median) unidentified; instead, only lower quantiles are identified. There is also a sample selection issue that only those who made it to a hospital alive are observed (54%); this becomes a problem if we wish to know their potential survival duration after hospitalization, if they had survived to a hospital contrary to the fact. The data set is limited in its covariates—only age is available—but includes the distance to the hospital, which plays an interesting role. Given that only age and distance are observed, it is likely that there are unobserved confounders. To account for them, a sensitivity analysis is conducted following pair matching. All estimators employed point to a clear winner and the sensitivity analysis indicates that the finding is fairly robust.     

Randomized response model versus mixture model in the crossover design for clinical trials

ABSTRACT

The clinical trials are usually designed with the implicit assumption that data analysis will occur only after the trial is completed. It is a challenging problem if the sponsor wishes to evaluate the drug efficacy in the middle of the study without breaking the randomization codes. In this article, the randomized response model and mixture model are introduced to analyze the data, masking the randomization codes of the crossover design. Given the probability of treatment sequence, the test of mixture model provides higher power than the test of randomized response model, which is inadequate in the example. The paired t-test has higher powers than both models if the randomization codes are broken. The sponsor may stop the trial early to claim the effectiveness of the study drug if the mixture model concludes a positive result.