Goppa code
Let be a monic polynomial over the finite field
and let
be a set of
elements of
such that
for
. The classical Goppa code
with Goppa polynomial
is the set of all words
in
for which
. Here
is to be interpreted as
where
is the unique polynomial of degree
degree
such that
.
The basic idea of this construction can be generalized as follows. Let be a non-singular projective curve (in the sense of algebraic geometry) defined over the finite field
. Let
be a set of rational points of
. Let
be the divisor
. Let
be another divisor with support disjoint from
. Let
be the linear system associated to the divisor
, i.e.
, where
is the set of non-zero rational functions on
and
is the divisor of zeros and poles defined by an
. The geometric Goppa code
of length
associated to the pair
is the image of the linear mapping
defined by
. If the supports of
and
are not disjoint, there is still a code associated to the pair
, but not a canonical one; however, all the codes so obtained are equivalent in a suitable sense. A second code associated to the pair
is the following. Let
be the vector space of rational differential forms
on
with
, with the zero form added. The second linear code
associated to the pair
is the image of the linear mapping
defined by
. This is the construction which more directly generalizes the classical Goppa codes mentioned above. The codes
and
are dual to each other. For fixed
and varying
the codes
and
yield the same family.
At present (1989) one approach to finding good codes is to find curves with large numbers of rational points compared to their genus. This brings in advanced algebraic geometry. Using Shimura curves (modular curves) one now can find a sequence of codes which beats the Gilbert–Varshamov bound for . For more details cf. [a4], [a7]–[a9].
References
[a1] | J.H. van Lint, "Introduction to coding theory" , Springer (1982) |
[a2] | A. Tietäväinen, "On the non-existence of perfect codes over finite fields" SIAM J. Appl. Math. , 24 (1973) pp. 88–96 |
[a3] | R.J. McEliece, E.R. Rodemich, H. Rumsey, L.R. Welch, "New upper bounds on the rate of a code via the Delsarte–MacWilliams inequalities" IEEE Trans. Inform. Theory , 23 (1977) pp. 157–166 |
[a4] | M.A. Tsfasman, S.G. Vladuts, T. Zink, "Modular curves, Shimura curves and Goppa codes, better than Varshamov–Gilbert bound" Math. Nachr. , 109 (1982) pp. 21–28 |
[a5] | P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) (Updated reprint) |
[a6] | R. Hill, "A first course in coding theory" , Clarendon Press (1986) |
[a7] | J.H. van Lint, G. van der Geer, "Introduction to coding theory and algebraic geometry" , Birkhäuser (1988) |
[a8] | V.D. Goppa, "Geometry and codes" , Kluwer (1988) |
[a9] | M.A. Tsfasman, S.G. Vlăduts, "Algebraic geometric codes" , Kluwer (1989) |
Goppa code. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Goppa_code&oldid=23847