Using an original mode of presentation, and emphasizing the
computational nature of the subject, this book explores a number of the
unsolved problems that still exist in coding theory. A well-established,
yet still highly relevant branch of mathematics, the theory of
error-correcting codes is concerned with reliably transmitting data over
a 'noisy' channel. Despite its frequent use in a range of contexts--the
first close-up pictures of the surface of Mars, taken by the NASA
spacecraft Mariner 9, were transmitted back to Earth using a Reed-Muller
code--the subject still contains interesting unsolved problems that have
resisted solution by some of the most prominent mathematicians of recent
decades.
Employing Sage--a free open-source mathematics software system--to
illustrate their ideas, the authors begin by providing background on
linear block codes and introducing some of the special families of codes
explored in later chapters, such as quadratic residue and
algebraic-geometric codes. Also surveyed is the theory that intersects
self-dual codes, lattices, and invariant theory, which leads to an
intriguing analogy between the Duursma zeta function and the zeta
function attached to an algebraic curve over a finite field. The authors
then examine a connection with the theory of block designs and the
Assmus-Mattson theorem. Further chapters scrutinize the knotty problem
of finding a non-trivial estimate for the number of solutions over a
finite field to a hyperelliptic polynomial equation of "small" degree,
as well as the best asymptotic bounds for a binary linear block code.
Some of the more mysterious aspects relating modular forms and
algebraic-geometric codes are also discussed.
Selected Unsolved Problems in Coding Theory is intended for graduate
students and researchers in algebraic coding theory, especially those
who are interested in finding some current unsolved problems.
Familiarity with concepts in algebra, number theory, and modular forms
is assumed. The work may be used as supplementary reading material in a
graduate course on coding theory or for self-study.
