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Implementation of an algorithm to compute the strong apparent distance of bivariate codes

Published in Journal of Physics: Conference Series, 2019

Abstract

The BCH bound is the oldest lower bound for the minimum distance of a cyclic code. The study of this bound and its generalizations are classical topics, which includes the study of the very well-known family of BCH codes. In 1970, P. Camion extended the notion of BCH bound to the family of abelian codes by introducing the apparent distance of polynomials. Camion showed that the minimum value of the apparent distance of certain polynomials associated to codewords is less than or equal to the minimum distance of the code. The mentioned minimum value is known as the apparent distance of an abelian code. In 2016, Bernal-Bueno-Simón introduced the notion of strong apparent distance of polynomials and hypermatrices and developed an algorithm to compute the minimum strong apparent distance of a hypermatrix based on g-orbits manipulations. In this work, we will present the implementation of an algorithm to compute the strong apparent distance of bivariate codes.

Recommended citation: Bueno-Carreño, D. H., & López, J. M. (2019, January). Implementation of an algorithm to compute the strong apparent distance of bivariate codes. In Journal of Physics: Conference Series (Vol. 1160, No. 1, p. 012012). IOP Publishing. https://iopscience.iop.org/article/10.1088/1742-6596/1160/1/012012/meta

Estimating Formation Mechanisms and Degree Distributions in Mixed Attachment Networks

Published in Journal of Physics A: Mathematical and Theoretical, 2019

Abstract

Our work introduces an approach for estimating the contribution of attachment mechanisms to the formation of growing networks. We present a generic model in which growth is driven by the continuous attachment of new nodes according to random and preferential linkage with a fixed probability. Past approaches apply likelihood analysis to estimate the probability of occurrence of each mechanism at a particular network instance, exploiting the concavity of the likelihood function at each point in time. However, the probability of connecting to existing nodes, and consequently the likelihood function itself, varies as networks grow. We establish conditions under which applying likelihood analysis guarantees the existence of a local maximum of the time-varying likelihood function and prove that an expectation maximization algorithm provides a convergent estimate. Furthermore, the in-degree distributions of the nodes in the growing networks are analytically characterized. Simulations show that, under the proposed conditions, expectation maximization and maximum-likelihood accurately estimate the actual contribution of each mechanism, and in-degree distributions converge to stationary distributions.

Recommended citation: Medina, J. A., Finke, J., & Rocha, C. (2019). Estimating formation mechanisms and degree distributions in mixed attachment networks. Journal of Physics A: Mathematical and Theoretical, 52(9), 095001. https://iopscience.iop.org/article/10.1088/1751-8121/aaffeb

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