Construction of Superimposed Codes Using Graphs and Galois Fields

University essay from Karlstads universitet/Institutionen för matematik och datavetenskap (from 2013)

Author: David Johansson; [2017]

Keywords: ;

Abstract: In this thesis some constructions of superimposed codes are presented. Many of the known nontrivial constructions arise from t−designs, and the constructions discussed in this thesis is also based on a block design idea. Superimposed codes are rather combinatorial in nature, so the connection to t−designs is not too surprising. What may be a little surprise, however, is the connection between superimposed codes and linear codes and Galois elds. Linear codes are quite intuitive and have nice properties, as is the case for Galois elds; combinatorial structures are quite often the contrary, not intuitive and quite dicult to understand. Because of this, it is interesting that a combinatorial structure like superimposed codes can be constructed from structures like linear codes and Galois elds. The main goal of this thesis is to present two possibly new approaches to construct superimposed codes. The constructions are described, but not proved to be correct. The rst construction presented is using graphs. In practice, this is not a good way to construct codes, since it requires the construction of a graph and nding certain cycles in the graph. It is still an interesting construction, however, since it provides a connection between constant weight codes and superimposed codes. Another construction is presented, one that seems much more useful when constructing codes. In [7] one particular superimposed code is constructed from a Galois eld. In this thesis we will see that this construction using Galois elds can be generalized.

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