id: 00500585
dt: j
an: 00500585
au: Munuera, Carlos; Pellikaan, Ruud
ti: Equality of geometric Goppa codes and equivalence of divisors.
so: J. Pure Appl. Algebra 90, No.3, 229-252 (1993).
py: 1993
pu: Elsevier Science B.V. (North-Holland), Amsterdam
la: EN
cc: 
ut: Goppa codes; equivalence of divisors; self-dual codes; algebraic- geometric
    codes
ci: 
li: doi:10.1016/0022-4049(93)90043-S
ab: The paper is concerned with algebraic-geometric codes (AG-codes). Let $G$
    and $H$ be divisors of the same degree $m$ on a curve of genus $g$. $D$
    is a divisor of degree $n$ of rational points on a curve
    $(D=\sum\sb{i=1}\sp{i=n} P\sb i)$ and the supports of $G$ and $H$ are
    disjoint from $D$. Let $L(K)$ be the set of rational functions $f$ on
    the curve having a divisor $\text{div} (f)\geq -K$, $K$ a divisor. The
    code $C(D,G)$ is defined as $C(D,G)= (f(P\sb 1),\dots, f(P\sb n))\mid
    f\in L(G)$. The main theorem of the paper is that if $2g-1<m< n-1$ then
    $C(D,G)= C(D,H)$ if and only if $G$ and $H$ are equivalent with respect
    to $D$ (i.e. $G= H+\text{div} (u)$ for some rational function $u$
    satisfying $u(P\sb i)=1$, for all $i=1,\dots, n)$. For $m=2g-1$ or
    $m=n-1$ there are some special cases in which $C(D,G)= C(D,H)$ while
    $G$ and $H$ are not equivalent themselves (but are equivalent up to a
    point). These cases are listed in Theorem 4.14, b, c, d and e. The main
    theorem leads in a natural way to a characterization of (formally)
    self-dual AG-codes $(n>2g+2)$: $C(D,G)$ is (formally) self- dual if and
    only if there exists a differential form $η$ with single poles at the
    $P\sb i$ such that $2G=D+ \text{div}(η)$ (in case of self duality
    $\text{res}\sb{P\sb i} (η)=1$). The paper concludes with counting the
    number of AG-codes on a curve arising from a divisor $D$ in terms of
    the generalized zeta-function of the curve with respect to $D$.
rv: H.J.Tiersma (Diemen)