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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Strongly Cancellative and Recovering Sets on Lattices. | Strongly Cancellative and Recovering Sets on Lattices Hoda Bidkhori ShinnYih Huang Department of Mathematics Department of Mathematics Northwestern University Illinois USA Yale University Connecticut USA hbidkho@ncsu.edu shinnyih.huang@yale.edu Submitted Dec 28 2010 Accepted Mar 24 2011 Published Mar 31 2011 Abstract We use information theory to study recovering sets R and strongly cancellative sets CL on different lattices. These sets are special classes of recovering pairs and cancellative sets previously discussed in the papers of Simonyi Frankl and Fiiredi. We mainly focus on the lattices Bn and Dk. Specifically we find upper bounds and constructions for the sets RBn CBn and CDk. 1 Introduction In this paper we study the strongly cancellative sets CL and recovering sets RL that are subsets of points in lattices L see Definition 2.1 and 2.2. On one hand the study of the former set is motivated by the work of Ahlswede Frankl and Furedi 8 and Fredman 7 . Specifically strongly cancellative sets are a special class of cancellative sets. On the other hand the study of recovering sets is prompted by the previous work of Simonyi 9 on recovering pairs. A recovering pair A B is an ordered pair of subsets A B of points in a lattice such that for any a a G A and b b G B we have the following a A b a A b a a a V b a V b b b . The paper of Korner and Olistky 5 shows that the upper bound of A B plays an important role in the zero-error information theory. Cohen gave an upper bound 3 for the size of A B on the Boolean lattice while Holzman and Korner improved the bound to 2.3264n afterward. Throughout this paper we study a special class of the recovering pairs Rl Rl which takes a single set RL. We call RL a recovering set. As Definition 2.1 and 2.2 shows a recovering set is a special case of a strongly cancellative set. Here we focus on the upper bounds and structures of these two sets by using Information Theory. This paper is organized as follows Section 2 presents the .
