Developments of the last few decades in digital communications have created a close link between mathematics and areas of computer science and electrical engineering. A collaboration between such areas now seems very natural due to problems which require deep knowledge and expertise in each area. A special role in such collaboration has played algebra and some of its branches such as algebraic geometry, computational algebra, group theory, etc. As a result of such cooperation now we have disciplines such as coding theory and cryptography which are considered a mix of mathematics, computer science, and electrical engineering.

Coding theory is one of the most important and direct applications of information theory. It is a branch of electrical engineering, digital communication, mathematics, and computer science designing efficient and reliable data transmission methods, so that redundancy in the data can be removed and errors induced by a noisy channel can be corrected. It started with Shannon, Hamming, and many others in the mid 20-th century and became one of the most active areas of research for most of the second half of the 20-th century. Algebraic coding theory was the main direction of coding theory, even though recently other ways of coding have been developed.

Cryptology, is the science of hiding information, and historically has received much attention from the public. As a science it also acquired a solid foundation in the second half of the 20-th century. It is a mixture of theoretical mathematics and computer science which focuses more in areas such as number theory, algebraic geometry, graph theory, algorithm analysis, etc. There have been many conferences and publications which have explored the common ground among such areas.

This volume comes out of the conference “New Challenges in Digital Communications”, Vlora, Albania, 2009. This Advanced Study Institute was funded by a NATO grant as a “Advanced Study Institute”. The conference focused precisely on connections between algebra, algebraic geometry, number theory, graph theory, and related areas of mathematics with coding theory and cryptography.

The conference which was organized at the University of Vlora, during April 27 - May 9, 2008 lasted two weeks and had lectures during the morning sessions and talks during late afternoons. There were over 130 participants in the institute from all over the world. The institute had 15 lecturers, namely:

A. Elezi, J. Gutierrez, W. C. Huffman, K. Magaard, J. Kozicki, G. Nebe, V. Pless, E. Previato, T. Shaska, F. Luca, S. Shpectorov, I. Shparlinski, A. Stein, V. Tonchev, V. Ustimenko, M. Ciperjani (invited speaker)

and the following additional speakers:

A. Kohnert, A. Gunther, A. Gomilko, C. Shor, S. Jakub Kotorowicz, A. Wrblewska, M. Wrobel, S. Chopuryan, L. Szalay, I. Siap, R. Sanjeeva, R. Scherbak, H.P.T. Viet, S. Chopuryean, G. Shaska, M. Ramosaco, N. Pjero

We want first to thank NATO, for providing the funds of the Institute. Without such support this institute would have not been possible.

We also want to thank the University of Vlora, which put all the time and effort in organizing such a big conference. Special thanks to the organizing staff especially Vice-Rector for Research and Development of the University of Vlora, Dr. Pranvera Resulaj, Gertian Balliu, Aulona Mustafaraj, Arjan Beqiri, Altin Mustafaraj, and all the students who volunteered with the conference. Special thanks to all the staff of the University of Vlora who were involved in all organizational tasks of the conference, especially the Department of Mathematics and the Department of Computer Science and Electrical Engineering, and the Vlora Conference Center at the University of Vlora. Further thanks to the Albanian coastguard of the city of Vlora for their help in making possible the boat trips for the conference participants.

We want to thank the Albanian Ministry of Science and Education for providing additional funding of such conference. Special thanks go to the Prime-minister’s advisor for education, Prof. Dr. M. Tafaj, and to the Vice-Minister for Education, Prof. Dr. Adriana Gjonaj for their support and encouragement during the conference.

Finally, we want to thank all the participants in the conference, especially Prof. Vera Pless, who came despite her health at the time, Prof. Shparlinski who came all the way from Sydney and all the other lecturers. Particular thanks to all the authors who contributed to this volume.

We hope the volume will be useful to mathematicians, computer scientists, and engineers who need to explore such ties between algebra and coding theory and cryptography. Most of the papers focus on coding theory and some others in cryptography. While such topics were the main focus of the conference, we had lectures which focused more on theoretical aspects such as computational group theory, computational algebraic geometry, theta functions, etc. Such areas have always provided a furtile ground in the area of communications. We hope that such collection of papers will serve the scientific community in mathematics, computer science, and electrical engineering and foster closer relations among such communities.

T. Shaska and E. Hasimaj

Vlora, Albania