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Perfect codes in the Lee and Chebyshev metrics and iterating Rédei functions

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Author(s):
Claudio Michael Qureshi Valdez
Total Authors: 1
Document type: Doctoral Thesis
Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática
Defense date:
Examining board members:
Michel Marie Deza; Marcelo Muniz Silva Alves; Emerson Luiz do Monte Carmelo; José Plínio de Oliveira Santos
Advisor: Sueli Irene Rodrigues Costa; Daniel Nelson Panario Rodriguez
Abstract

The content of this thesis is inserted in two very active research areas: the theory of error correcting codes and dynamical systems over finite fields. To approach problems in both topics we introduce a type of finite sequence called v-series. A metric is introduced in the set of such sequences inducing a poset structure used to determine all possible abelian group structures represented by perfect codes in the Chebyshev metric. Moreover, each v-serie is associated with a rooted tree, which has an important role in results related to the cycle structure of iterating Rédei functions. Regarding the theory of error correcting codes, we study perfect codes in the Lee metric and Chebyshev metric (corresponding to the lp metric for p=1 and p=infinity, respectively). The main results here are related to the description of n-dimensional q-ary codes with packing radius e which are perfect in these metrics, obtaining their generator matrices and their classification up to isometry and up to isomorphism. Several constructions of perfect codes in the Chebyshev metric are given and interesting families of such codes are presented. Regarding dynamical system over finite fields we focus on iterating Rédei functions, where our main result is a structural theorem, which allows us to extend several results on Rédei functions. The above theorem can also be applied to other maps, allowing simpler proofs of some known results related to the number of components, the number of periodic points and the expected value for the period and preperiod for iterating exponentiations over finite fields (AU)