Teaching Discrete Mathematics with Computer Algebra:  
A New Approach to Modern Algebra and Error-Correcting Codes

Kenneth Hulth (huke@konrad.hj.se) and Igor Gachkov (gaig@konrad.hj.se)

Abstract for the session _Computer Algebra in Education_ at IMACS-ACA'97

An attempt to introduce Computer Algebra in the course program of the School of
Engineering, Jonkoping University, Sweden, was made three years ago, as the
course Discrete Mathematics with Computer Algebra was given for the first time.
The response from the students was very encouraging and indicated that a
conceptually new course structure with hands-on sessions, using the powerful
tools of MATHEMATICA, could make an advanced theory available for students with
a minimal background in mathematics.

In developing the course modules, the authors used the MATHEMATICA package
Discrete Mathematics, and a set of laboratory sessions on an individual basis
was worked out.

The content of courses in discrete mathematics usually encompass some theory on
algebraic structures, and often, as an application, one or two chapters on how
to correct errors due to noise in communication channels.  The demand of
knowledge in this field has in recent years increased tremendously, due to the
now-a-days daily use of plastic (type credit-) cards in money-machines, in
payment operations, keys of different kinds, as well as in CD-Players, CD-ROM
storage in computers, FAX-machines etc, which all use error-correcting codes
based on modern (abstract) algebra.

Thus, the wide spread use of electronic circuits with error-correcting
functions (practically every electronic circuit produced today has some sort of
error-correcting capacity) gave the authors the idea to develop a special
course module focusing on coding theory for engineers, not having deep
mathematical background, but whose educational program is strongly connected
with the construction, implementation and/or technical maintenance of the
mentioned equipment, i.e., engineers in telecommunication, electronics or
informatics.

Due to the necessity of using advanced mathematical theories, like results on
algebraic structures including extensions of finite fields and linear spaces
over Galois fields, courses in coding theory in general have to be allocated on
post-graduate level.  But having the possibilities of computer algebra at their
disposal, the authors got the idea to develop a non-standard,
methodical-oriented course where hands-on sessions could add substantial
understanding in the introduction of the mentioned mathematical concepts.
Consequently, a package in MATHEMATICA on Coding Theory was developed at the
School of Engineering, and the first course was given autumn 1994 for students
of advanced under-graduate level.  The hands-on sessions, worked out as a
concept with introductory and problem-solving parts, usually repeated and
sometimes also demonstrated the theory from the preceding lectures in another
fashion.

The first results from the course and the structure of the package Coding
Theory were presented at the IMACS-ACA conference in Albuquerque 1995.

In the autumn semester 1996 the course was given for the third time.  With the
experience from the previous courses, with an extended package of commands, and
also using hints and suggestions from the students, the course has been further
developed.  Especially the previously not possible (due to complicated
mathematical aspects) parts of the codes BCH, Reed-Solomon have now been
included in the package.  The beautiful Error-Trapping decoding is demonstrated
both analytically and as from a practical point of view (with construction of
schemes and demonstration of the process of decoding).

Apart from mentioned course modules, the students at School of Engineering can
choose to write a (limited) scientific project in discrete mathematics using
computer algebra as the final project of the undergraduate program in
Jonkoping.  Also from this field the authors have positive experiences.