| Embedding unique customer identification as a watermark into multimedia data is called digital fingerprinting ,which makes it possible to trace the illegal usages of pirates. The serious problem in digital fingerprint is collusion attack. The key problem is to against the collusion attack by using codes with traceability properties. In this thesis, we research codes with traceability property (ω-FP code,ω-SFP code,ω-IPP code andω-TA code).In Chapter 2, firstly, we study the combinatorial properties ofω-IPP codes by giving its a tight characterization, and obtain necessary and sufficient conditions for a code to be aω-IPP code. So we make the combinatorial properties ofω-IPP for caseω= 2 andω≥3 unified. Furthermore, we give the nonconstructive existence results for linear 2-IPP codes and optimalω-IPP codes with length s(ω).In Chapter 3, we study the graphic properties ofω-IPP codes with lengthω+ 1. We obtain necessary and sufficient conditions for a code with lengthω+ 1 to be aω-IPP code. Using the graphic properties we solve the existence problem of optimalω-IPP codes with lengthω+ 1 completely. Firstly, we study the properties and structures of minimal q-ary (ω+1)-colorω-IPP code graphs, and give the classifies of them. Then, by direct construction we obtain the lower bound of the size of optimalω-IPP code of lengthω+ 1, and with the nonlinear programing we give the upper bound of this size. In the end, we give an efficient algorithm to give explicit constructions of optimalω-IPP codes of lengthω+ 1.In Chapter 4 section 1, using Chinese Reminder Theory and resolvable incomplete block design, we give explicit constructions ofω-FP code andω-SFP code, we also give inductive constructions forω-FP code. In section 2, using the inclusion relations of set system and liear subspaces, we give two classes of codes with traceabilω, respectively. Furthermore, we give necessary and sufficient conditions and sufficient conditions for these codes to beω-IPP codes andω-TA codes, respectively.
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