Bounds on edit metric codes with combinatorial DNA constraints

The design of a large and reliable DNA codeword library is a key problem in DNA based computing. DNA codes, namely sets of fixed length edit metric codewords over the alphabet {A, C, G, T}, satisfy certain combinatorial constraints with respect to biological and chemical restrictions of DNA strands. The primary constraints that we consider are the reverse--complement constraint and the fixed GC--content constraint, as well as the basic edit distance constraint between codewords. We focus on exploring the theory underlying DNA codes and discuss several approaches to searching for optimal DNA codes. We use Conway's lexicode algorithm and an exhaustive search algorithm to produce provably optimal DNA codes for codes with small parameter values. And a genetic algorithm is proposed to search for some sub--optimal DNA codes with relatively large parameter values, where we can consider their sizes as reasonable lower bounds of DNA codes. Furthermore, we provide tables of bounds on sizes of DNA codes with length from 1 to 9 and minimum distance from 1 to 9.

Generator Matrix Based Search for Extremal Self-Dual Binary Error-Correcting Codes

Self-dual doubly even linear binary error-correcting codes, often referred to as Type II codes, are codes closely related to many combinatorial structures such as 5-designs. Extremal codes are codes that have the largest possible minimum distance for a given length and dimension. The existence of an extremal (72,36,16) Type II code is still open. Previous results show that the automorphism group of a putative code C with the aforementioned properties has order 5 or dividing 24. In this work, we present a method and the results of an exhaustive search showing that such a code C cannot admit an automorphism group Z6. In addition, we present so far unpublished construction of the extended Golay code by P. Becker. We generalize the notion and provide example of another Type II code that can be obtained in this fashion. Consequently, we relate Becker's construction to the construction of binary Type II codes from codes over GF(2^r) via the Gray map.

Construction of I-Deletion-Correcting Ternary Codes

Finding large deletion correcting codes is an important issue in coding theory. Many researchers have studied this topic over the years. Varshamov and Tenegolts constructed the Varshamov-Tenengolts codes (VT codes) and Levenshtein showed the Varshamov-Tenengolts codes are perfect binary one-deletion correcting codes in 1992. Tenegolts constructed T codes to handle the non-binary cases. However the T codes are neither optimal nor perfect, which means some progress can be established. Latterly, Bours showed that perfect deletion-correcting codes have a close relationship with design theory. By this approach, Wang and Yin constructed perfect 5-deletion correcting codes of length 7 for large alphabet size. For our research, we focus on how to extend or combinatorially construct large codes with longer length, few deletions and small but non-binary alphabet especially ternary. After a brief study, we discovered some properties of T codes and produced some large codes by 3 different ways of extending some existing good codes.

The FCLT with Dependent Errors: an Helicopter Tour of the Quality of the Approximation

This note investigates the adequacy of the finite-sample approximation provided by the Functional Central Limit Theorem (FCLT) when the errors are allowed to be dependent. We compare the distribution of the scaled partial sums of some data with the distribution of the Wiener process to which it converges. Our setup is purposely very simple in that it considers data generated from an ARMA(1,1) process. Yet, this is sufficient to bring out interesting conclusions about the particular elements which cause the approximations to be inadequate in even quite large sample sizes.

Testing Mean-Variance Efficiency in CAPM with Possibly Non-Gaussian Errors : An Exact Simulation-Based Approach

In this paper we propose exact likelihood-based mean-variance efficiency tests of the market portfolio in the context of Capital Asset Pricing Model (CAPM), allowing for a wide class of error distributions which include normality as a special case. These tests are developed in the frame-work of multivariate linear regressions (MLR). It is well known however that despite their simple statistical structure, standard asymptotically justified MLR-based tests are unreliable. In financial econometrics, exact tests have been proposed for a few specific hypotheses [Jobson and Korkie (Journal of Financial Economics, 1982), MacKinlay (Journal of Financial Economics, 1987), Gib-bons, Ross and Shanken (Econometrica, 1989), Zhou (Journal of Finance 1993)], most of which depend on normality. For the gaussian model, our tests correspond to Gibbons, Ross and Shanken’s mean-variance efficiency tests. In non-gaussian contexts, we reconsider mean-variance efficiency tests allowing for multivariate Student-t and gaussian mixture errors. Our framework allows to cast more evidence on whether the normality assumption is too restrictive when testing the CAPM. We also propose exact multivariate diagnostic checks (including tests for multivariate GARCH and mul-tivariate generalization of the well known variance ratio tests) and goodness of fit tests as well as a set estimate for the intervening nuisance parameters. Our results [over five-year subperiods] show the following: (i) multivariate normality is rejected in most subperiods, (ii) residual checks reveal no significant departures from the multivariate i.i.d. assumption, and (iii) mean-variance efficiency tests of the market portfolio is not rejected as frequently once it is allowed for the possibility of non-normal errors.

Correcting the Errors : A Note on Volatility Forecast Evaluation Based on High-Frequency Data and Realized Volatilities

This note develops general model-free adjustment procedures for the calculation of unbiased volatility loss functions based on practically feasible realized volatility benchmarks. The procedures, which exploit the recent asymptotic distributional results in Barndorff-Nielsen and Shephard (2002a), are both easy to implement and highly accurate in empirically realistic situations. On properly accounting for the measurement errors in the volatility forecast evaluations reported in Andersen, Bollerslev, Diebold and Labys (2003), the adjustments result in markedly higher estimates for the true degree of return-volatility predictability.

Lecons de droit criminel: contenant l'explication compléte des codes pénal et d'instruction criminelle

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