are designed to transmit information (a 'signal') concisely and reliably,
using sequences of code-words.
To minimise the effect of errors or noise in transmission,
the code-words must be clearly different from each other.
For example, one code
comprises the following sixteen binary code-words of length seven:
which mutually differ in at least 3 places,
if each received code-word contains at most 1 error,
then the correct signal can still be extracted.
Thus if the first code-word
1101000 is sent,
but becomes corrupted en route and is received as
(with one error),
it will still be correctly decoded as
(the closest valid code-word ).
Coding theory can identify
points at which to evaluate a function
for efficient numerical integration.
For example, a parity bit can be added to each of the above code-words,
giving 16 code-words of length 8,
of which the following 14 have four '0's and four '1's:
from which to construct nice configurations of points
around the origin in 8-dimensional space,
Note also that the matrix
the 14 points (lying in a 7-d subspace)
obtained by replacing each '0' by '-1', or
the 16*14 = 224 points obtained by replacing each '1' independently
by '+1' or '-1'
(and by adding the 16 points with one coordinate = +2 or -2,
and the other seven all 0, we obtain Gosset's 8-dimensional
polytope, an important configuration in
that also arises as the minimal vectors in the E8
or equivalently as the centres of the 240 'spheres' touching a given
one in the densest possible 8-dimensional
is the incidence matrix of a regular
also represents the balanced incomplete block
ABD BCE CDF DEG AEF BFG ACG,
and (by adjoining a row and a column of '1's)
produces a Hadamard matrix, important throughout
The symmetries of structures like these are best studied using
MacWilliams & Sloane (1977)
The coding theory 'Bible' (or at least the Old Testament).
van Lint (1992)
a contender for the title of 'New Testament'.
Assmus & Key (1992)
Cameron & van Lint (1991)
Conway & Sloane (1992)
I have always sought to be understood,
and when my words were garbled by critics or colleagues,
I considered it no fault of theirs but my own,
because I had not been clear enough to be comprehended.
Henri Matisse, Testimonial.
We exchanged many frank words in our respective languages.
Beyond the Fringe.
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