The thesis is divided into two independent parts. The first part
examines the main types of fingerprinting codes under four descendant
models, while the second investigates the combinatorial object called
a honeycomb array.
Digital fingerprinting is a technique that is used to protect
intellectual rights by preventing illegal redistribution of digital
data (films, music, software, etc.). This technique is facilitated by
the collection of codes called fingerprinting codes. The thesis
focuses on the following four fingerprinting codes: traceability, IPP,
secure frameproof and frameproof. These codes are studied under four
models, namely narrow-sense, expanded narrow-sense, wide-sense and
expanded wide-sense. These models refer to the ability of malicious
users (traitors) to produce the fingerprint in the illegal copy. In
particular, following an idea of Boneh and Shaw, it is shown that
there only exist trivial wide-sense traceability and IPP codes. In the
matter of widesense frameproof codes, enhancing the relation between
these codes and Sperner families first introduced by Stinson and Wei,
we improve their upper bound on the size of this type of
fingerprinting codes. The last two results are original and we regard
the latter to be the main original contribution of this part of the
thesis.
A honeycomb array of radius r is a set of n = 2r + 1 dots placed on
the hexagonal grid in such a way that the distance of every dot from a
fixed cell, the centre, is at most r. It is also required that in each
column and in each diagonal only one dot occurs and that the vector
differences between all pairs of dots are distinct. In the thesis it
is proved that honeycomb arrays can only be constructed using Costas
arrays, which are configurations of dots in the square grid similar to
honeycomb arrays. Using the existing Costas array database, all
honeycomb arrays with r <= 14 are determined, and two new arrays of
radius 7 are presented.