| 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. We begin by associating to each code an edge colored graph. Then a code has the IPP if and only if its associated graph has certain structural conditions. We study the underlying structure of graphs associated with IPP codes of maximum size. Using this approach, we present explicit construction of classes of graphs associated with IPP codes of length 3, which gives a lower bound on F(3,q). By further investigating the structure of graphs associated with IPP codes of length 3, we show that there exist maximum IPP codes of length 3 whose associated graphs have good structural properties. Using such structural properties, we are able to convert the problem of deciding F(3,q) to a nonlinear programming problem. Based on our nonlinear programming formulation, we give an algorithm which determines F(3,q) numerically for each q ≥ 15. We also describe how to construct maximum IPP codes from optimal solutions to the nonlinear programming problem, and show that such IPP codes possess good tracing capabilities. Using techniques from nonlinear programming, we prove a precise formula for F(3,q) when q ≥ 15.; Our approach may be used to improve bounds on F(2 k+1,q). For example, we characterize the associated graphs of maximum IPP codes of length 5, and obtain bounds on F(5, q). |
