Maximum Codes with the Identifiable Parent Property
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We study codes that have identifiable parent property. Such codes are called IPP codes. Research on IPP codes is motivated by design of schemes that protect against piracy of digital products. Construction and decoding of maximum IPP codes have been studied in rich literature. General bounds on F(n,q), the maximum size of IPP codes of length n over an alphabet with q elements, have been obtained through the use of techniques from graph theory and combinatorial design. Improved bounds on F(3,q) and F(4,q) are obtained. Probabilistic techniques are also used to prove the existence of certain IPP codes. We prove a precise formula for F(3,q), construct maximum IPP codes with size F(3,q), and give an efficient decoding algorithm for such codes. The main techniques used in this thesis are from graph theory and nonlinear optimization. Our approach may be used to improve bounds on F(2k+1, q). For example, we characterize the associated graphs of maximum IPP codes of length 5, and obtain bounds on F(5,q).