Conference rusure::math

    Algebraic coding theory is a particular branch of coding theory.  Sort
    of like French food is food or VAX is a computer.  Nowadays, just about
    all the work done in coding theory is done is about algebraic code. 
    Information theory, although related, is different beast.  Information
    theory deals with upper/lower bounds of the codes, but gives few clue as
    to how to design good codes.  Sort of like the homotopy theory and the
    homology theory in algebraic topology.
    
    Eugene

    >Sort of like the homotopy theory and the homology theory in algebraic
    >topology.
    Iam afraid even to ask about this, Eugene :-)

    Homotopy and Homology were invented by Henri Poincare.

    Consider a torus, or the complex plane with poles or singularities.
    Two curves are homotopic if they can be deformed into each
    other.  On the torus, there are two simple inequivalent closed curves
    that can't be deformed to each other or shrunk to a point - any
    other closed curve can be "generated" by combinations of these.
    Thus, these curves generate the fundamental group for the
    torus.  On the complex plane, you have curves that don't
    cross branch points etc.  The idea came from complex analysis.
    It works for higher dimensional manifolds too.

    Returning to the torus, consider triangulating the surface of it
    (like you might do to draw computer graphics.)

    A point is a zero dimensional simplex, a line is one dimensional
    (having points as boundaries), a triangle is 3 dimensional
    and has lines as boundaries, a tetrahedron is 4 dimensional
    and has triangles as boundaries and so on. (Simplicial homology)

    A k-chain is a set of k-simplices, and the boundary operator returns
    the (k-1) boundary chain of a set of simpices.  Now a cycle is a
    closed chain whose boundary is zero, but is not necessarily a boundary
    - that is it does not necessarily arise in applying the boundary operator
    to some higher dimensional chain.  Note that the boundary of a boundary
    is always zero!

    These chains form groups, and it is natural to look at the
    quotient group of closed chains (whose boundary is zero) modulo the
    closed chains which are also boundaries (ie which came from applying
    the boundary operator to a higher dimensional chain).

    This quotient group is the so-called simplicial homology group.

    Symbolically, H = Z / B.

    Any closed chain which is a boundary is homologous to zero.

    As you can see there are some connections between homotopy
    and homology but they talk about different things.

    Note that there is a dual to homology called cohomology.  This arises
    when you look at vector fields (and more generally differential forms)
    on manifolds; topology comes up when you ask which forms are exact,
    arising from exterior differentiation of a lower form, and which
    are closed, having a vanishing exterior derivative.  (A gravitational
    field is exact, it's a gradient, magnetic fields are not.)

    DeRham showed that homology and cohomology are equivalent, they
    provide the same information.

    This is from memory, I'm no expert; a really nice introduction
    to the ideas would come from a book on Riemann surfaces like Springers
    book.  It's really neat stuff, if only I had the time to learn more
    about it.

    - Jim

>> 2B::MATH Note 1371.3
>>    A point is a zero dimensional simplex, a line is one dimensional
>>    (having points as boundaries), a triangle is 3 dimensional
>>    and has lines as boundaries, a tetrahedron is 4 dimensional
>>    and has triangles as boundaries and so on. (Simplicial homology)

	You added a dimension to the triangle and tetrahedron.

	Dan

   Re the base question:

   Coding theory itself doesn't emphasise the algebraic aspect of the
   subject - for example, a code could conceivably be defined which
   required a massive lookup table for the translation.  Using
   abstract algebra and number theory one can add a great deal of
   structure to the codes, making them practical.  There is probably
   almost no coding theory as such which doesn't have *some* algebra
   lurking in there.

   I only learned something about abstract algebra after getting interested
   in coding - the subject (algebra) wouldn't have appealed to me at all
   before seeing it used.  Nonetheless, I'm glad I did learn a little
   about it, since I see the ideas used in other branches of mathematics
   too.  Like most engineers, I have to be pulled kicking and screaming...

   - Jim

	As I recall from my coding theory classes, block codes are best analyzed
	by algebraic means, because all possible words of a given length form
	a group with interesting subgroups, including the subgroup of all
	possible code words (i.e., those that are legitimate transmissions, they
	do not contain an error).

	However, convolutional codes, such as the Viterbi encoder/decoder, take
	a continuous input stream, and encode it.  There is no natural block, 
	or _word_ structure due to the code itself.  I may be remembering wrong,
	but I don't think there is much you can do algebraically with these 
	codes.

	Incidentally, the CD format uses a Reed-Solomon block code to correct
	errors, which is why a CD will still play even if pretty scratched up.

	-fab