Some structural properties of convolutional codes over rings

Convolutional codes over rings have been motivated by phase-modulated signals. Some structural properties of the generator matrices of such codes are presented. Successively stronger notions of the invertibility of generator matrices are studied, and a new condition for a convolutional code over a ring to be systematic is given and shown to be equivalent to a condition given by Massey and Mittelholzer (1990). It is shown that a generator matrix that can be decomposed into a direct sum is basic, minimal, and noncatastrophic if and only if all generator matrices for the constituent codes are basic, minimal, and noncatastrophic, respectively. It is also shown that if a systematic generator matrix can be decomposed into a direct sum, then all generator matrices of the constituent codes are systematic, but that the converse does not hold. Some results on convolutional codes over Z(pe) are obtaine

Convolutional codes under control theory point of view. Analysis of output-observability

In this work we make a detailed look at the algebraic structure of convolutional codes using techniques of linear systems theory. The connection between these concepts help to better understand the properties of convo- lutional codes, in particular the concepts of controllability and observability of linear systems can be translated into the context of convolutional codes relating these properties with the noncatastrophicity of the codes. We examine the output-observability property and we give conditions for this property.Postprint (published version

Learning Multimodal Graph-to-Graph Translation for Molecular Optimization

We view molecular optimization as a graph-to-graph translation problem. The goal is to learn to map from one molecular graph to another with better properties based on an available corpus of paired molecules. Since molecules can be optimized in different ways, there are multiple viable translations for each input graph. A key challenge is therefore to model diverse translation outputs. Our primary contributions include a junction tree encoder-decoder for learning diverse graph translations along with a novel adversarial training method for aligning distributions of molecules. Diverse output distributions in our model are explicitly realized by low-dimensional latent vectors that modulate the translation process. We evaluate our model on multiple molecular optimization tasks and show that our model outperforms previous state-of-the-art baselines

Noncatastrophic convolutional codes over a finite ring

Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized as polynomial left prime matrices. In this paper, we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in the finite ring Zpr. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring Zpr. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated Zpr-modules. We introduce the notion of b-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on Zp is.The second and third authors were supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciancia e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. Diego Napp is partially supported by Ministerio de Ciencia e Innovación via the grant with ref. PID2019-108668GB-I00