This book provides a first course on lattices - mathematical objects pertaining to the realm of discrete geometry, which are of interest to mathematicians for their structure and, at the same time, are used by electrical and computer engineers working on coding theory and cryptography. The book presents both fundamental concepts and a wealth of applications, including coding and transmission over Gaussian channels, techniques for obtaining lattices from finite prime fields and quadratic fields, constructions of spherical codes, and hard lattice problems used in cryptography. The topics selected are covered in a level of detail not usually found in reference books. As the range of applications of lattices continues to grow, this work will appeal to mathematicians, electrical and computer engineers, and graduate or advanced undergraduate in these fields.

The most commonly deployed multi-storage device systems are RAID housed in a single computing unit. The idea of distributing data across multiple disks has been naturally extended to multiple storage nodes which are interconnected over a network and are called Networked Distributed Storage Systems (NDSS). The simplest coding techniques based on replication are often used to ensure redundancy in these systems, but given the sheer volume of data that needs to be stored and the overheads of replication, other coding techniques are being developed. Coding Techniques for Repairability in Networked Distributed Storage Systems (NDSS) surveys coding techniques for NDSS, which aim at achieving (1) fault tolerance efficiently and (2) good repairability characteristics to replenish the lost redundancy, and ensure data durability over time. This is a vibrant research and this book is the first overview that presents the background required to understand the problems as well as covering the most important techniques currently being developed. Coding Techniques for Repairability in Networked Distributed Storage Systems is essential reading for all researchers and engineers involved in designing and researching computer storage systems.

Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space-Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion, requires that the difference of any two distinct codewords has full rank. Extensive work has been done on Space-Time coding, aiming to attain fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space-Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to improve the design of good codes. Cyclic Division Algebras: A Tool for Space-Time Coding provides a tutorial introduction to the algebraic tools involved in the design of codes based on division algebras. The different design criteria involved are illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property and the diversity multiplexing tradeoff. Finally complete mathematical background underlying the construction of the Golden code and the other Perfect Space-Time block codes is given. Cyclic Division Algebras: A Tool for Space-Time Coding is for students, researchers and professionals working on wireless communication systems.

Algebraic number theory is gaining an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems. Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been developed in the last ten years and many explicit code constructions based on algebraic number theory are now available. Algebraic Number Theory and Code Design for Rayleigh Fading Channels provides an overview of algebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to algebraic number theory. The basic facts of this mathematical field are illustrated by many examples and by the use of computer algebra freeware in order to make it more accessible to a large audience. This makes the book suitable for use by students and researchers in both mathematics and communications.