\input amstex
\documentstyle{amsppt}
\NoBlackBoxes


\normalbaselineskip=1.6\normalbaselineskip\normalbaselines
\magnification=1200




\def\min{\mathop{\text{min}}}
\def\rk{\mathop{\text{rk}}}
\def\Gm{{\Bbb G}_m}
\def\ra{\rightarrow}
\def\Z{{\Bbb Z}}
\def\topn{\buildrel{p^n}\over\to}
\def\Q{{\Bbb Q}}
\def\C{{\Bbb C}}
\def\F{{\Bbb F}}
\def\zmp{\Z/p\Z}
\def\O{{\Cal O}}
\def\isom{\cong}
\def\Ext{\mathop{\hbox{Ext}}}
\def\Qp{{\Bbb Q}_p}
\def\Qpb{\bar{\Bbb Q}_p}
\def\Cp{{\Bbb C}_p}
\def\P{{\Bbb P}}
\def\Hom{\mathop{\rm Hom}\nolimits}
\def \H{{\Cal H}}
\def \X{{\Cal X}}
\def\Spec{\mathop{\rm Spec}\nolimits}
\def \bs{\bigskip}
\def\Gal{\mathop{\rm Gal}\nolimits}
\def\End{\mathop{\rm End}\nolimits}
\def\limproj{\mathop{\oalign{\hfil$\rm lim$\hfil\cr
 $\longleftarrow$\cr}}}
\def \L{\Lambda}
\def\bk{\bar{k}}
\def \ra{\rightarrow}
\def \op{\frac{1}{p}}
\def \st{\stackrel}
\def \da{\downarrow}
\def \R{{\bf Rings}}
\def \d{\delta}
\def \s{\sigma}
\def \a{\alpha}
\def \b{\beta}
\def \x{\chi}
\def \t{\tau}
\def \z{\zeta}
\def \con{\equiv }
\def \e{\epsilon}
\def \bC{\bar{C}}
\def \bX{\bar{X}}
\def \k{\kappa}
\def \bF{{\bar{\Bbb F}}_p}


\def \m{{\eufm m}}
\def \spec{\operatorname{Spec}}
\def \supp{\operatorname{supp}}
\def \cl{{\Cal L}}
\def \co{{\Cal O}}
\def \cz{{\Cal Z}}
\def \cp{{\Cal P}}
\def \ck{{\Cal K}}
\def \ce{{\Cal E}}
\def \cf{{\Cal F}}
\def \x{{\bold x}}
\def \y{{\bold y}}
\def \real{\operatorname{Re}}
\def \ct{{\Cal T}}


\centerline{\bf Euclidean weights of codes from elliptic curves over
rings }
\medskip
\centerline{\bf Jos\'e Felipe Voloch and Judy L.~Walker}
\bigskip

{\bf 1. Introduction}

\bigskip

The purpose of this paper is to construct certain error-correcting
codes over finite rings and estimate their parameters. For
this purpose, we need to develop some tools; notably an estimate for
certain exponential sums and some results on canonical lifts of
elliptic curves. These results may be of independent interest.

A code is a subset of $A^n$, where $A$ is a finite set (called the
alphabet). Usually $A$ is just the field of two elements and, in this
case, one speaks of binary codes.  Such codes are used in applications
where one transmits information through noisy channels. By building
redundancy into the code, transmitted messages can be recovered at the
receiving end.  A code has parameters that measure its efficiency and
error-correcting capability. For various reasons one often restricts
attention to linear codes, which are linear subspaces of $A^n$ when
$A$ is a field. However, there are non-linear binary codes (such as
the Nordstrom-Robinson, Kerdock, and Preparata codes) that outperform
linear codes for certain parameters. These codes have remained somewhat
mysterious until recently when Hammons, et al.~([HKCSS]) discovered
that one can obtain these codes from linear codes over rings
(i.e. submodules of $A^n$, $A$ a ring) via the Gray mapping, which
we recall below.

In a different vein, over the last decade there has been a lot of
interest in linear codes coming from algebraic curves over finite
fields.  The construction of such codes was first proposed by Goppa in
[G]; see [St] or [TV] for instance.  In [TVZ], it is proven that for
$q \ge 49$ a square, there exist sequences of codes over the finite
field with $q$ elements which give asymptotically the best known
linear codes over these fields.  The second author has extended
Goppa's construction to curves over local Artinian rings and shown,
for instance, that the Nordstrom-Robinson code can be obtained from
her construction followed the Gray mapping; see [W1] and [W2].
While most of the parameters for these new codes were estimated in the
above papers, the crucial parameter needed to describe the
error-correcting capability of the images of these codes under the
Gray mapping was still lacking.  In this paper we consider the second
author's construction in the special case of elliptic curves which are
defined over finite local rings and which are the canonical lifts of
their reductions. (See section 4 for more about canonical lifts.) For these
codes, the missing parameter can be estimated and we do so.

Another application of our construction is to obtain low-correlation
sequences suitable for use in code-division multiple access (CDMA)
schemes, which are used when multiple users need to share a common
communication channel, such as in the case of cellular telephones. We
will use our results to obtain such sequences. In a way, our results
are the analogues for elliptic curves of the results of Kumar et
al.~([KHC]), which can be viewed as being for the multiplicative
group. Since we can work with any ordinary elliptic curve over a
finite field, our results are more flexible.

This paper is organized as follows. In section 2 we recall the main
results of [W1] on the construction of codes from curves over rings
and review the definitions pertaining to error-correcting codes. We
also set the stage in this section for the results we need. In section
3 we prove a general estimate for certain exponential sums along
curves. This result extends a number of recent results but,
paradoxically, is based on an old paper of H.~L.~Schmid. In section 4
we prove a number of results about canonical lifts of elliptic
curves. Finally, in section 5, we put everything together, obtaining
our main results and their applications.

\bigskip

{\bf 2. Algebraic geometric codes over rings }

\bigskip

In \cite{W1}, the idea of algebraic geometric codes over rings other
than fields is introduced, and foundational results about these codes
are proven.  In \cite{W2}, the methods of \cite{W1} are used to
explicitly construct the $\Z/4$-version of the Nordstrom-Robinson code
as an algebraic geometric code.  In order to construct other codes
over $\Z/4$ with good nonlinear binary shadows, we must first
investigate the Lee and Euclidean weights of these codes.  In this
section, we recall the definitions and some results from \cite{W1}
and explain how the Lee and Euclidean weights of algebraic geometric
codes over rings are related to exponential sums.

Let $A$ be a local Artinian ring with maximal ideal $\m$.  We assume
that the field $A/\m$ is finite; say $A/\m = \F_q$.  Let ${\bold X}$
be a curve over $A$, that is, a connected irreducible scheme over
$\spec A$ which is smooth of relative dimension one.  Let ${\bold X}
\times_{\spec A} \spec \F_q = X \subset {\bold X}$ be the fiber of
${\bold X}$ over the closed point of $\spec A$.  We assume $X$ is
absolutely irreducible, so that it is the type of curve on which
algebraic geometric codes over $\F_q$ are defined.  Let $\cz = \{Z_1,
\dots, Z_n\}$ be a set of $A$-points on ${\bold X}$ with distinct
specializations $P_1, \dots, P_n$ in $X$.  Let $G$ be a (Cartier)
divisor on ${\bold X}$ such that no $P_i$ is in the support of $G$,
and let $\cl = \co_{\bold X}(G)$ be the corresponding line bundle.
For each $i$, $\Gamma(Z_i, \cl|_{Z_i}) \simeq A$, and thinking of
elements of $\Gamma({\bold X}, \cl)$ as rational functions on ${\bold
X}$, we may think of the composition $\Gamma({\bold X}, \cl) \to
\Gamma(Z_i, \cl|_{Z_i}) \to A$ as evaluation of these functions at
$Z_i$.  Summing over all $i$, we have a map $\gamma: \Gamma({\bold X},
\cl) \to \bigoplus \Gamma(Z_i, \cl|_{Z_i}) \to A^n$, given by $f \mapsto
(f(Z_1), \dots, f(Z_n))$.

\definition{Definition 2.1} Let $A$, ${\bold X}$, $\cz$, $\cl$, and
$\gamma$ be as above.  Define $C_A({\bold X},\cz,\cl)$ to be the image
of $\gamma$.  $C_A({\bold X}, \cz, \cl)$ is called the algebraic
geometric code over $A$ associated to ${\bold X}$, $\cz$, and $\cl$.
\enddefinition

The following theorem summarizes some of the main results of
\cite{W1}.

\proclaim{Theorem 2.2} Let ${\bold X}$, $\cl$, and $\cz = \{Z_1,
\dots, Z_n\}$ be as above.  Let $g$ denote the genus of ${\bold X}$,
and suppose $2g-2 < \deg \cl < n$.  Set $C = C({\bold X}, \cz, \cl)$.
Then $C$ is a linear code of length $n$ over $A$, and is free as an
$A$-module.  The dimension (rank) of $C$ is $k = \deg \cl + 1 - g$,
and the minimum Hamming distance of $C$ is at least $n - \deg \cl$.
Further, under the additional assumption that $A$ is Gorenstein, the
class of algebraic geometric codes is closed under taking duals.  In
particular, there exists a line bundle $\ce$ such that $C^\perp =
C({\bold X}, \cz, \ce)$.
\endproclaim

\remark{Remark 2.3} The dimension and minimum Hamming distance
computations are consequences of the Riemann-Roch Theorem, and it is
here that the assumption $2g-2 < \deg \cl < n$ is used.  The duality
result follows from a generalized version of the Residue Theorem which
holds for Gorenstein rings.  See \cite{W1} for details.
\endremark

The following result will be useful in section 5.

\proclaim{Lemma 2.4}  Let $A$, $\bold X$, and $\Cal L$ be as
in Theorem 2.2.  Let $\bold K$ be the total quotient ring of rational
functions on $\bold X$, and let $\bold L = \Gamma(\bold X, \Cal L)$.
Define $p^{-1} \bold L = \{ \bold g \in \bold K \, \mid \, p
\bold g \in \bold L\}$.  Then $p^{-1} \bold L = \bold L
+ \ker(\beta)$, where $\beta$ is the map $\bold K \to \bold K$ given
by multiplication by $p$.
\endproclaim

\demo{Proof} Consider the map of the short exact sequence $0 \to \bold
L \to \bold K \to \bold K/\bold L \to 0$ to itself given by
multiplication by $p$.  By the Snake Lemma, the kernels
and cokernels fit into an exact sequence.  But it is shown in
\cite{W1} that $\bold L \otimes_A A/m = \Gamma(X, \Cal L')$, where
$\Cal L'$ is the pullback of $\Cal L$ to $X$.  This means that the
cokernels form a short exact sequence by themselves, so the kernels
must also.  To set up notation, let $\gamma: \bold K/\bold L \to \bold
K/\bold L$ be multiplication by $p$, let $\pi_0$ be the surjection
$\ker(\beta) \to \ker(\gamma)$, and let $\pi$ be the surjection $\bold
K \to \bold K/\bold L$.  With this notation, $p^{-1} \bold L =
\ker(\pi \beta) = \ker(\gamma \pi)$.

Let $\bold g \in p^{-1} \bold L$.  Then $\pi(\bold g) \in
\ker(\gamma)$.  Since $\pi_0$ is surjective, there is some $\bold g'
\in \ker(\beta)$ with $\pi_0(\bold g') = \pi(\bold g)$.  But then
$\pi(\bold g - \bold g') = 0$, so $\bold g - \bold g' \in \bold L$.
In other words, there is some $\bold f' \in \bold L$ with $\bold g =
\bold f' + \bold g'$, which is precisely what we needed to show.
\enddemo


For applications, one is usually concerned with constructing codes
over $\Z/4$, or more generally, over rings of the form $\Z/p^l$, where
$p$ is prime and $l \ge 1$.  We can use algebraic geometry to
construct such codes in two different ways.  First, we can simply set
$A = \Z/p^l$ in the definition of algebraic geometric codes above.
Alternatively, we can construct an algebraic geometric code over
$GR(p^l, m)$ and look at the associated trace code over $\Z/p^l$.
Here, $GR(p^l, m)$ denotes the degree $m \ge 1$ Galois extension of
$\Z/p^l$ (see, for example, \cite{KHC} for details).  It is easily
seen that such a ring is isomorphic to the ring of length $l$ Witt
vectors over the field $\F_{p^m}$, and this representation is used in
sections 3 and 4 below.  In particular, there is a trace map
$T:GR(p^l, m) \to \Z/p^l$, and by the trace code of a code, we mean
the code obtained by applying this trace map coordinatewise to the
codewords.

The Gray map allows us to construct (non-linear) binary codes from
codes over $\Z/4$ and is defined as follows. Consider the map
$\varphi: \Z/4 \to \F_2^2$ defined by $\varphi(0)=(0,0), \varphi(1) = (0,1),
\varphi(2)=(1,1),\varphi(3)=(1,0)$. Now we define a map, again denoted by
$\varphi: \Z/4^n \to \F_2^{2n}$, by applying the previous $\varphi$ to each
coordinate.

For linear codes over rings of the form $\Z/p^l$, it is
often either the Euclidean or Lee weight rather than the Hamming
weight which is of interest.  In particular, when $p^l = 4$, the
Euclidean and Lee weights are closely related, and the Lee weight
gives the Hamming weight of the associated nonlinear binary code.

We begin by defining Euclidean weights.  We identify an element $x$ of
the cyclic group $\Z/p^l$ with the corresponding $p^l$th root
of unity via the map
$$
x \to e_{p^l}(x) := e^{2 \pi i x / {p^l}}.
$$

\definition{Definition 2.5} The Euclidean distance between $x$ and
$y$ is the distance $d_E(x,y)$ in the complex plane between the points
$e_{p^l}(x)$ and $e_{p^l}(y)$, and the Euclidean weight of $x$ is the
distance $w_E(x)$ between $e_{p^l}(x)$ and $e_{p^l}(0) = 1$.
\enddefinition
We have
$$
w_E(x) = \sqrt{\sin^2 \left({2 \pi x \over p^l}\right) + (1 - \cos
\left({2 \pi x \over p^l}\right))^2} = \sqrt{2 - 2 \cos \left({2 \pi x
\over p^l}\right)}. 
$$

In fact, it is usually the square of the Euclidean weight in which one is
interested.  This is given by $w_E^2 (x) = 2 - 2 \cos ({2 \pi x \over
{p^l}})$.  For vectors $\x = (x_1, \dots, x_n)$ and $\y = (y_1, \dots,
y_n)$ over $\Z/{p^l}$, we define
$$
\align
d_E^2(\x,\y) &= \sum_{j = 1}^n d_E^2(x_j,y_j) \\
\intertext{and} 
w_E^2(\x) &= \sum_{j = 1}^n w_E^2(x_j)
\endalign
$$

For example, the squared Euclidean weight of the all-one vector in
$({\Bbb Z}/p^l)^n$ is $2n(1 - \cos({2 \pi/p^l}))$.  Using the Taylor
expansion of cosine, we get that this is at least $4n {\pi^2 \over
p^{2l}}(1 + {\pi^2 \over 3 p^{2l}})$.  Further, any other nonzero
constant vector in $({\Bbb Z/p^l})^n$ has squared Euclidean weight at
least this.

For general vectors, since $\cos ({2 \pi x \over {p^l}})=
\real\left(e_{p^l}(x)\right)$, we have  
$$
\align
w_E^2(\x) &= \sum_{j = 1}^n\left(2 -
2\real\left(e_{p^l}(x_j)\right)\right)\\ 
& = 2n - 2 \real \sum_{j = 1}^n e_{p^l}(x_j) \\ 
& \geq 2n - 2 \left|\sum_{j = 1}^n e_{p^l}(x_j)\right|.
\endalign
$$
Hence, to find a lower bound on the minimum Euclidean weight of a
linear code over $\Z/{p^l}$, it is enough to find an upper bound on
the modulus of the exponential sum
$$
\sum_{j = 1}^n e_{p^l}(x_j).
$$

Now consider the case ${p^l} = 4$.  Then $e_4(0) = 1$, $e_4(1) = i$,
$e_4(2) = -1$, and $e_4(3) = -i$.  Hence $w_E^2(0) = 0$, $w_E^2(1) =
w_E^2(3) = 2$, and $w_E^2(2) = 4$.  Since the Lee weight is defined by
$w_L(0) = 0$, $w_L(1) = w_L(3) = 1$, and $w_L(2) = 2$, we have 
$$
w_L(x) = \frac12 w_E^2(x)
$$
for any $x \in \Z/4$.  From this we see that the Euclidean weight of a
codeword over $\Z/4$ is twice the Hamming weight of the binary
codeword obtained by applying the Gray map.  Notice that the Lee
weight of a constant vector in $({\Bbb Z}/4)^n$ is either $0$, $n$, or
$2n$. 

Finally, let $C$ be an algebraic geometric code over $GR(p^l, m)$, and
let $T: GR(p^l, m) \to \Z/p^l$ denote the trace map as before.  We are
interested in the minimum Euclidean weight of $T(C)$, the trace code
of $C$, which is a linear code over $\Z/p^l$.  Codewords in $T(C)$ are
of the form $(T(f(Z_1)), \dots, T(f(Z_n)))$, where $f$ is a rational
function on some curve ${\bold X}$ defined over $GR(p^l,m)$ and $Z_1,
\dots, Z_n$ are $GR(p^l,m)$-points on ${\bold X}$. From the argument
above, to find a lower bound for the minimum Euclidean weight of
$T(C)$ it suffices to find an upper bound on the modulus of
$$
\sum_{j=1}^n e_{p^l}(T(f(Z_j))) = \sum_{j=1}^n e^{2 \pi i T(f(Z_j))/p^l}.
$$

We investigate this sum in sections 3 and 4 below.

\bigskip

{\bf 3. Exponential sums}

\bigskip


In this section we will give estimates for some kinds of exponential
sums along curves. The approach follows the classical method of
relating the exponential sum to the sum of the reciprocals of the
zeros of an $L$-function and applying the Riemann hypothesis.  
Of course the abstract set-up is well-known in even greater
generality (see, for example, [D]), but it is the calculation of the
degree of the $L$-function that requires working out.  We will be
dealing with characters of order $p^l$, where $p$ is the
characteristic. In this case, the degree of the $L$-function was
computed by Schmid [S1], [S2]. 

Let $X$ be a curve over the finite field $\F_q$, where $q = p^m$ with
$p$ prime. Denote by $K = \F_q(X)$ the function field of $X$. Let
$f_0,\ldots,f_{l-1} \in K$ and consider the Witt vector ${\bold f} =
(f_0,\ldots,f_{l-1}) \in W_l(K)$.  Let $X_0$ be the maximal affine
open subvariety of $X$ where $f_0,\ldots,f_{l-1}$ do not have poles
and let $P \in X_0(\F_q)$.  We can then consider the Witt vector
${\bold f}(P) = (f_0(P),\ldots,f_{l-1}(P)) \in W_l(\F_q)$. Letting
$T:W_l(\F_q) \to W_l(\F_p) \isom \Z/p^l\Z$ denote the trace map as in
section 2, we can consider the exponential sum
$$
S_{{\bold f},\F_q} = \sum_{P \in X_0(\F_q)} e^{2\pi i T({\bold
f}(P))/p^l}.
$$



\proclaim{Theorem 3.1} With notation as above, assume $X \setminus
X_0$ consists of the points above the valuations $v_1,\ldots,v_s$ of
$K$.  Let $g$ be the
genus of $X$, $n_{ij} = -v_j(f_i), i= 0,\ldots, l-1,j=1,\ldots,s$, 
and assume that
$\bold f$ is not of the form $\bold f = F(\bold g) - \bold g + \bold
c$ for any $\bold g \in W_l(K)$ and $\bold c \in W_l(\F_q)$, where $F$
denotes the additive endomorphism on $W_l(K)$ given by $F(g_0, g_1,
\dots, g_{l-1}) = (g_0^p, g_1^p, \dots, g_{l-1}^p)$.  Then $|S_{{\bold
f},\F_q}| \le Bq^{1/2}$, where 
$$B \le 2g-1 + 
\sum_{j=1}^s \max \{ p^{l-1-i}n_{ij} \mid 0 \le i \le l-1 \}\deg v_j.$$
\endproclaim


\demo{Proof} As mentioned above, the essential steps are done in [S1],
[S2], but we will repeat them here for the reader's
convenience. Consider the Artin-Schreier-Witt extension $Y: F({\bold
y})-{\bold y} = {\bold f}$ of $X$, which is a cover of $X$ with Galois
group contained in $\Z/p^l\Z$. Assume first it is a geometric cover,
i.e., that there is no constant field extension, and that it has
positive degree.  If $\chi$ is a character of the Galois group then we
can form the (Artin) $L$-function $L(X,\chi,t)$.  As long as $\chi \ne
1$, $L(X, \chi, t)$ is a polynomial in $t$ of a certain degree
$B_{\chi}$ satisfying 
$$B_{\chi} \le 2g - 1 + \sum_{j=1}^s \max\{p^{l-1-i} n_{ij} \mid
0 \le i \le l-1 \}\deg v_j,$$
with equality holding if $\chi$ is injective (see
[S1], [S2], and especially [S2], Satz 8).  When $\chi = 1$,
$L(X,\chi,t)$ is the zeta function of $X$. Taking the product over all
characters $\chi$ of the Galois group of $Y/X$, $\prod L(X,\chi,t)$ is the zeta
function of $Y$.  From this our bound will follow since, by the
general theory (e.g. [S2](2.7)), $S_{{\bold f},\F_q} = \sum_{P \in
X_0(\F_q)} \chi(P)$ for a character $\chi$, where $\chi(P)$ means
$\chi$ evaluated at the Frobenius substitution of $P$.  Also, $\sum_{P
\in X_0(\F_q)} \chi(P)$ equals the negative of the sum of the
reciprocals of the roots of $L(X,\chi,t)$ and these roots have
absolute value $q^{-1/2}$ by the Riemann hypothesis.

We now treat the case where $Y$ is not necessarily a geometric cover
of $X$.  Let $L = K(\bold y)$ so $L/K$ is cyclic. Now if $\F_q$ is not
algebraically closed in $L$, then $L$ contains $k = \F_{q^p}$.  Set $M
= K k \subset L$, so that $[M:K] = [k:\F_q] = p$.  Since $L$ is
cyclic, the intermediate field extension of degree $p$ over $K$ is
unique, so we have $M = K(y_0)$.  Thus $K(y_0)/K$ is a constant field
extension, which implies that $f_0 = g^p - g + a$ for some $g \in K$
and $a \in \F_q$.  Letting ${\bold g} = (g,0,\ldots,0) \in W_l(K)$ and
${\bold a} = (a,0,\ldots,0) \in W_l(\F_q)$, we can find ${\bold h} \in
W_{l-1}(K)$ such that ${\bold f} = F({\bold g}) - {\bold g} + {\bold
a} + p{\bold h}$. Then
$$
S_{{\bold f},\F_q} = e^{2\pi i T({\bold a})}\sum_{P \in X_0(\F_q)}
e^{2\pi i T({\bold h}(P))/p^{l-1}}.
$$
Notice that if $\bold h = F(\bold k) - \bold k + \bold d$ with $\bold
k \in W_{l-1}(K)$ and $\bold d \in W_{l-1}(\F_q)$, then $\bold f =
F(\bold g) - \bold g + \bold a + p (F(\bold k) - \bold k + \bold d) =
F(\bold g + p \bold k) - (\bold g + p \bold k) + (\bold a + \bold d)$,
which contradicts the hypothesis of the theorem.  Therefore, we can
assume by induction on $l$ that $|S_{\bold f, \F_q}| \le C q^{1/2}$,
with 
$$C \le 2g - 1 + 
\sum_{j=1}^s \max\{p^{l-2-i} m_{ij} \mid 0 \le i \le l-2\}\deg v_j,$$
where $m_{ij} = -v_j(h_i)$.  Further, a computation gives $m_i \le
\max \{p^{i-j} n_j \mid 0 \le j \le i+1\}$, so in fact 
$$C \le 2g - 1 +
\sum_{j=1}^s \max\{p^{l-2-i} n_{ij} \mid 0 \le i \le l-1\}\deg v_j,$$ 
and from this the theorem follows.
\enddemo




\remark{Remark 3.2} There has been some recent interest on exponential
sums of the kind considered in the above theorem, in the case of
$\P^1$, see Kumar et al.~([KHC]) and Li ([L]).  These authors use
elements of $W_l(\F_q)[x]$ instead of $W_l(\F_q[x])$, to form their
exponential sums. The latter is a bigger ring but, for exponential
sums, it doesn't matter. W. Li has informed us that their results
can be deduced from the above theorem.
In any rate, we can recover directly the applications in [KHC]
to low-correlation sequences by following the same procedure of
section 5, replacing the elliptic curve by the multiplicative group.

\bigskip


{\bf 4. Canonical liftings}

\bigskip


Let $E$ be an ordinary elliptic curve defined over a finite field
$\F_q$.  Then $E$ has a canonical lifting to an elliptic curve over
$W(\F_q)$ for which the Frobenius of $E$ also lifts. This is a special
case of the Serre-Tate theory (see [LST] or [K]). If ${\bold E}$
denotes the lift of $E$ to $W(\F_q)$, there is also an injective
homomorphism $\t: E({\bar \F_q}) \to {\bold E}(W({\bar \F_q}))$
(analogous to the Teichm\"uller lift for $\Gm$), compatible with the
action of Frobenius, which we will call the elliptic Teichm\"uller
lift (see [Bu2]).  In fact, a characterization of the canonical lift
is the existence of such a homomorphism. We will recall its
construction below.

On the other hand ${\bold E}$ has a Greenberg transform $G({\bold E})$
which is an infinite-dimensional scheme over $\F_q$, together with a
map $\gamma: {\bold E}(W({\bar \F_q})) \to G({\bold E})({\bar
\F_q})$. Our purpose is to compute the degrees of the Witt coordinate
functions of $\gamma \circ \t$.  By [Bu2] we know that $\gamma \circ
\t$ actually corresponds to a section of the canonical morphism of
$\F_q$-schemes $G({\bold E}) \to E$. By abuse of language, we will
often identify ${\bold E}$ and $G({\bold E})$ in what follows.

\proclaim{Theorem 4.1} Let $E$ be an ordinary elliptic curve
defined over a finite field $\F_q$ and ${\bold E}$ the canonical lift
of $E$ to $W(\F_q)$. Let ${\bold G}$ be an effective Cartier divisor
on $\bold E$ and $G$ its restriction to $E$. Let ${\bold f}$ be a
global section of $\co_{\bold E}({\bold G})$.  Then for $P \in E, P \notin
\supp {\bold G}$,
${\bold f}(\t(P)) = (f_0(P),f_1(P),\ldots)$ as a Witt vector, where
$f_i$ is a global section of $\co_E((2p)^i G)$ for $i = 0, 1,
\dots$.
\endproclaim

\demo{Proof} A procedure for computing the $f_i$'s is given in [Bu1],
Lemmas 2.6 and 2.7. First one computes the $p$-jet coordinates, $g_i$
say, which are reduction modulo $p$ of $\d^i{\bold f}$, where $\d u =
(u \circ \phi -u^p)/p$ is a $p$-derivation on the structure sheaf of
${\bold E}/W(\F_q)$ and $\phi$ is the lift of Frobenius on ${\bold
E}/W(\F_q)$, as follows from Lemma 2.7 of [Bu1] together with [Bu2].
As $\phi$ is an isogeny of degree $p$, it follows that $\deg \d u$ is
at most $2p \deg u$ and therefore, by induction $\deg g_i \le
(2p)^i \deg G$.  Now, from Lemma 2.6 of [Bu1], $f_i +
P_i(f_0,\ldots,f_{i-1})=g_i$, for universal polynomials $P_i$ computed
in the proof there. On the other hand, it is clear that the $f_i$ are
regular away from $\supp G$ and thus so are the $g_i$.  Using the proof of
Lemma 2.6 of [Bu1], one can show that $P_n(zf_0, z^p f_1, \dots,
z^{p^{n-1}} f_{n-1}) = z^{p^n} P_n(f_0, f_1, \dots, f_{n-1})$ for $z
\in K$, so that monomials of $P_n(f_0, f_1, \dots, f_{n-1})$ are of
the form $f_0^{i_0} f_1^{i_1} \dots f_{n-1}^{i_{n-1}}$, with $i_0 + p
i_1 + \dots p^{n-1} i_{n-1} = p^n$.  This implies that $\deg(P_n(f_0,
f_1, \dots, f_{n-1})) \le \max\{p^{n-i} \deg f_i | 0 \le i \le n-1\}$.
A straightforward induction argument now gives $\deg f_i \le (2p)^i\deg G$
and the theorem then follows.
\enddemo

Suppose that ${\bold x},{\bold y}$ are coordinates
of a Weierstrass equation for ${\bold E}$. The above proof then
produces functions $x_0,x_1,\ldots,y_0,y_1,\ldots$ on $E$ such that
$x_0,y_0$ are the coordinates of the reduced Weierstrass equation for
$E$ and 
$$
\t((x_0,y_0)) = ((x_0,x_1,\ldots),(y_0,y_1,\ldots)).
$$  

By reducing modulo $p^l$, we can consider the canonical lift of $E$ to
$W_l(\F_q)$.  The following proposition gives us tools to help
explicitly calculate this in the important case $l=2$.

\proclaim{Proposition 4.2} Let $k$ be a perfect field of
characteristic $p > 0$ and $E/k$ an ordinary elliptic curve.  If
${\bold E}$ is the canonical lift of $E$ to $W_2(k)$, then $\deg x_1 <
3p, \deg y_1 < 4p$. Conversely, let ${\bold E}$ be any elliptic curve
defined over $W_2(k)$ with reduction $E$. Assume that the projection
given by reduction from $G({\bold E})$ to $E$ admits a section $\t$ in
the category of $k$-schemes over $E \setminus \{O\}$ (where $O$ is the
origin for the group law on $E$) given by $(x_0,y_0) \mapsto (\bold
x,\bold y)=((x_0,x_1),(y_0,y_1))$ where $x_1,y_1$ are regular away
from $O$ and satisfy $\deg x_1 < 3p, \deg y_1 < 4p$. Then $\t$ is
regular at $O$, ${\bold E}$ is the the canonical lift of $E$ and $\t$
is the elliptic Teichm\"uller lift.
\endproclaim

\demo{Proof} Theorem 4.1 gives that $\deg x_1 \le 4p$ but since both 
${\bold x}\circ \phi$ and ${\bold x}^p$ have a pole at the origin of
${\bold E}$, we actually get $\deg x_1 < 4p$.  Here again, $\phi$
denotes the lift of Frobenius on $\bold E$.  Consider the differential
$\phi^*(d{\bold x}/{\bold y})/p$. As shown by Mazur [M], this is a
well-defined, holomorphic differential on ${\bold E}$ and its
reduction modulo $p$, $\omega$ say, depends only on $dx/y$. Moreover
$C(\omega) = dx/y$, where $C$ is the Cartier operator.  Hence $\omega
= A^{-1}dx/y$, where $A$ is the Hasse invariant of $E$.  On the other
hand, from the proof of Theorem 4.1,
$$
{1 \over p}\phi^*({d{\bold x} \over {\bold y}}) = {1 \over
p}{{d({\bold x}^p + px_1)} \over {{\bold y}^p + py_1}}=
{x_0^{p-1}dx_0+dx_1 \over y_0^p}.
$$ 
This gives $dx_1/dx_0 = A^{-1}y_0^{p-1} - x_0^{p-1}$, which is a
polynomial in $x_0$ of degree $3(p-1)/2$. Since $\deg x_1 < 4p$, this
determines $x_1$ up to a linear combination of $1$ and $x_0^p$ and
thus $x_1$ is a polynomial in $x_0$ of degree $(3p-1)/2$, hence $\deg
x_1 < 3p$.  Examining the Weierstrass equation for ${\bold E}$ gives
the bound for $y_1$.

To show the converse, first we need to show that $\t$ is regular at
$O$ and $\t(O) = {\bold O}$, where $\bold O$ is the origin for the
group law on $\bold E$.  It is enough to show that $\bold x/\bold y$
is regular at $\bold O$ and that $\bold x/\bold y({\bold O})=0$.  A
computation gives $\bold x/\bold y=(x_0/y_0, x_1/y_0^p -
y_1x_0^p/y_0^{2p})$ and both $x_1/y_0^p$ and $y_1x_0^p/y_0^{2p}$
vanish at $O$, since $\deg x_1 < 3p, \deg y_1 < 4p$.

Fix $P_0 \in E, P_0 \ne O$ and consider $f(P) =
\t(P+P_0)-\t(P)-\t(P_0)$.  So $f$ is a morphism from $E$ to
$\ker(G({\bold E}) \to E)$.  However $E$ is projective and
$\ker(G({\bold E}) \to E)$ is affine, so $f$ is constant. But
$f(O)={\bold O}$ so $f={\bold O}$ and $\t$ is a homomorphism. From the
definition of the canonical lift, this forces ${\bold E}$ to be the
canonical lift of $E$. Finally if $\t'$ is the elliptic Teichm\"uller
lift then $\t-\t'$ is a morphism from $E$ to $\ker(G({\bold E}) \to
E)$.  And $(\t-\t')(O)={\bold O}$, so by the same argument as above,
$\t=\t'$.
\enddemo

\remark{Remark 4.3} Applying Proposition 4.2 we can check the following 
examples of canonical lifts. 
If ${\bold E}$ is given by $\bold y^2 +\bold x \bold y =
\bold x^3 - \bold x^2 -2\bold x -1$ over $W(\bar{\F}_2)$, then it is
the canonical lift of its reduction modulo two. A computation gives
$x_1=1, y_1= x_0^2(1+y_0)$.  More generally, if $k$ is a field of
characteristic $2$ and $a \in k, a \ne 0$, consider the elliptic curve
$E/k$ given by $y_0^2 + x_0y_0 = x_0^3 + a$. Its canonical lift to
$W_2(k)$ is $\bold y^2 + \bold x\bold y = \bold x^3 + (a,a^2)$ and the
elliptic Teichm\"uller lift is given by $\bold x=(x_0,a), \bold y =
(y_0, (x_0^2+x_0)y_0 + x_0^3 + ax_0^2 + a)$.  The canonical lift to
$W_2(k)$ of $y_0^2= x_0^3 + x_0^2 + a$ over a field $k$ of
characteristic three is $\bold y^2 = \bold x^3 + \bold x^2 + (a,0)$
and the elliptic Teichm\"uller lift is $\bold x=(x_0, x_0^4 +
(1-a)x_0^3 + ax_0 - a^2), \bold y = (y_0, x_0^2y_0)$.  For another
example, $\bold y^2=\bold x^3+\bold x$ is the canonical lift of its
special fiber in characteristic five and the elliptic Teichm\"uller
lift to $W_2$ is given by $(x_0,y_0) \mapsto (\bold x, \bold y) =
((x_0,x_1),(y_0,y_1))$ where $x_1= 4x_0^7 + x_0^3, y_1= y_0(x_0^8 +
2x_0^6 + 2x_0^4 + x_0^2 +3)$.  These examples show that the bounds in
Proposition 4.2 on $\deg x_1$ and $\deg y_1$ cannot be any smaller.
\endremark

\proclaim{Corollary 4.4} Notation as in the theorem.  If ${\bold f}
\circ \tau \ne F({\bold g}) - {\bold g} + {\bold c}$, for any ${\bold
g} \in W_l(K), {\bold c} \in W_l(\F_q)$, then
$$
| \sum_{P \in E_0(\F_q)} e^{2\pi i T({\bold f} 
(\tau(P)))/p^l} | \le ((2p)^{l-1}\deg G + 1)q^{1/2}.
$$
\endproclaim

\demo{Proof} Combine Theorem 3.1 with Theorem 4.1.
\enddemo

\remark{Remark 4.5} If ${\bold E}/W_2(k)$ is the lift of an elliptic curve
$E/k$ given by a Weierstrass equation with coordinates ${\bold x},{\bold y}$, 
the global sections of $\co_{\bold E}(r {\bold O})$ are of the form
$A+B{\bold y}$, where $A,B$ are polynomials in ${\bold x}$ of degrees at most
$[r/2],[(r-3)/2]$ respectively. It follows from the examples in remark
4.3 that, in the situation of Theorem 4.1 in characteristic two, we
have $\deg f_1 \le 2r + 1$, which improves slightly on the bound $4r$
coming from Theorem 4.1. Correspondingly, we can improve the bound in
the above corollary to $(2r+2)q^{1/2}$.
\endremark

\remark{Remark 4.6} It follows from [Bu1], Propositions 1.7 and 1.8,
that, if there is a section of the reduction map from a projective curve
${\bold X}/W(\F_q)$ to its special fibre $X/\F_q$ then $X$
is of genus zero or one. If the genus is zero then the section exists.
If the genus is one and $X$ is ordinary  the section exists if and
only if ${\bold X}$ is the canonical lift of $X$, hence the
restrictions in the above result. It is possible to obtain sections of
reductions of affine curves over Witt vectors of finite length, but
the degree of the sections grow much faster than that given by the
above theorem, so the bounds are correspondingly worse.
\endremark

\bigskip

{\bf 5. Applications}

\bigskip


We now return to the study of Euclidean weights of algebraic geometric
codes defined using elliptic curves over Galois rings.  In order to
apply the results of sections 3 and 4 above, we make a few extra
assumptions. Let ${\bold E}$ be an elliptic curve defined over the
Galois ring $A = GR(p^l,m) = W_l(\F_q)$, where $q = p^m$.  Let $E$ be
the fiber of ${\bold E}$ over the closed point of $\spec A$, i.e., the
reduction modulo $p$ of ${\bold E}$.  Then $E$ is an elliptic curve
over $\F_q$.  We now make the additional assumptions that $E$ is
ordinary and that ${\bold E}$ is the canonical lift of $E$ to $A$ (by
which we mean the reduction modulo $p^l$ of the canonical lift of $E$
to $W(\F_q)$, as discussed in section 4.)

Let $Z_0$ be an $A$-point on ${\bold E}$ and let $P_0$ be the
corresponding $\F_q$-rational point of $E$.  Set $E_0 =
E\setminus\{P_0\}$. Let $\{P_1, \dots, P_n\} = E_0(\F_q)$, and let
$\cz = \{Z_1, \dots, Z_n\}$, where $Z_i = \tau(P_i)$ for $i = 1,
\dots, n$ and $\tau$ is the canonical lifting of points from section
4.  Let $r \ge 1$ and set $\cl = \co_{\bold E} (r Z_0)$.

\proclaim{Theorem 5.1} Let $A$, ${\bold E}$, $\cz$, $\cl$
be as above. Let $C = C_A({\bold E}, \cz, \cl)$,
and let $T:A \to W_l(\F_p) = \Z/p^l$ denote the absolute trace map.
Then the minimum squared Euclidean weight of $T(C)$
satisfies
$$
w_E^2(T(C)) \ge \min \{2n - ((2p)^{l-1} r + 1) p^\frac{m}2, 4n
{\pi^2 \over p^{2l}}(1 + {\pi^2 \over 3 p^{2l}})\}.
$$
\endproclaim

\demo{Proof} Simply choose the group law for ${\bold E}$ so that $Z_0$
is the origin and hence $P_0$ is the origin for $E$.  Then this
theorem is a direct consequence of Corollary 4.4. Indeed, we need only
to bound the Euclidean weight of the images of those $\bold f$ of the
form $F({\bold g)} - {\bold g} + {\bold c}$, where ${\bold g} \in
W_l(K), {\bold c} \in W_l(\F_q)$. In this case, $(T(\bold f(P_1)),
\dots, T(\bold f(P_n)))$ is a constant vector so the bound follows
from the discussion in section 2.

\enddemo

 
\proclaim{Corollary 5.2} Use the same notation as above, but now
assume that $p = l = 2$, so that $T(C)$ is a code over $\Z/4$.  Then
the minimum Lee weight of $T(C)$, and hence the minimum Hamming weight
of $\varphi(T(C))$, where $\varphi$ is the Gray map described in the
introduction, satisfies
$$
w_L(T(C)) = w_H(\varphi(T(C))) \ge \min\{n - (2r+2) 2^{m-3 \over 2}, n\}.
$$
\endproclaim

\demo{Proof} Immediate from Remark 4.5, since $w_H(\varphi(T(C))) =
w_L(T(C)) = \frac12 w_E^2(T(C))$, as explained in section 2.
\enddemo

Our methods can also be used to construct low-correlation sequences
for use in CDMA (Code Division Multiple Access) communications
systems, which are used in applications such as cellular telephones.
For details on this, the reader is referred to \cite{KHC} and the
references therein.  We include here only a very brief overview of the
basic idea.  We consider infinite sequences of period $n$ with symbols
in $\Z/p^l$.  The idea is to form a large family of such sequences
which are pairwise cyclically distinct (i.e.  no sequence is a shift
of any other) and which have small correlation.  Here the correlation
$c_{st}(\Delta)$ between sequences $s = \{s(j)\}$ and $t = \{t(j)\}$
of period $n$ for shift $\Delta$ is
$$
c_{st}(\Delta) = \left| \sum_{j=0}^{n-1} e^{2 \pi i (s(j+\Delta) - t(j))
\over p^l}\right|
$$
One measures whether or not a family of sequences is good by
considering the maximum correlation parameter
$$
C_{\max} = \max\{c_{st}(\Delta) \, | \, s, t \in \cf \text{ and either
$s \ne t$ or $\Delta \ne 0$}\}.
$$
In applications, the large family size allows for a large number of
users and a small correlation parameter translates to little
interference from one user to another.

A family of sequences can be constructed as follows.  Take ${\bold E}$
as above and assume that $E(\F_q)$ is cyclic of order $n$, and let $P$
be its generator. Let ${\bold G}$ be a Cartier divisor on ${\bold
E}/W_l(\F_q)$ consisting of the Galois orbit of the lift of a point of
$E$ having degree $r$.  In particular, $\bold G$ is a sum of distinct
points and so is its reduction $G$. We will assume that $(r,n)=1$ and
the following lemma will be useful in the proof of Theorem 5.4 below.

\proclaim{Lemma 5.3} Let $E/\F_q$ be an elliptic curve with $n$
rational points and let $r$ be an integer, $(r,n)=1$. For any point
of $E$ of degree $r$ over $\F_q$ and for any $\sigma$ in the Galois
group of $\F_q(P)/\F_q$,
we have that $P^{\sigma}-P \notin E(\F_q) \setminus \{ O \}$.
\endproclaim

\demo{Proof} Let $\sigma$ be of order $d|r$. If 
$P^{\sigma}-P = P_0 \in E(\F_q)$, then 
$$
O = P^{\sigma^d} - P = \sum_{i=1}^d P^{\sigma^i} - P^{\sigma^{i-1}} =
\sum_{i=1}^d (P^\sigma - P)^{\sigma^{i-1}} = 
dP_0.
$$
However, since $d$ is coprime to $n$, this implies $P_0 =O$, proving the
lemma.
\enddemo

For each class in $\Gamma({\bold E},\co_{\bold E}({\bold G}))/W_l(\F_q)$, 
choose a representative ${\bold f}$ in 
$\Gamma({\bold E},\co_{\bold E}({\bold G}))$ and
define the sequence $s_{\bold f}(j)= T({\bold f}(\tau(jP_1)))$. 

\proclaim{Theorem 5.4} Let $\cf$ be the family of sequences defined above.
Then the maximum correlation parameter $C_{\max}$ of $(\cf)$ satisfies
$$
C_{\max} \le 1 + ((2p)^{l-1}2\deg {\bold G} + 1) p^\frac{m}2.
$$
Furthermore, if $\deg {\bold G}$ and the above bound for 
$C_{\max}$ are both less than $n$, 
then $|\cf| = q^{l(\deg {\bold G} - 1)}$.
\endproclaim

\demo{Proof} The correlation of $s_{\bold f}$ and $s_{\bold g}$ 
for shift $\Delta$ is
$$
| \sum_{j=1}^n e^{2\pi i 
T({\bold f}(\tau(jP_1))-{\bold g}(\tau((j+\Delta)P_1)))/p^l} |. 
$$
Consider the automorphism $\alpha$ of ${\bold E}$ given by translation
by $\tau(\Delta P_1)$. Let ${\bold h} = {\bold f} - {\bold
g}\circ\alpha$.  The above sum is then the kind of sum considered in
Corollary 4.4, say, but with ${\bold G}$ replaced by ${\bold
G}+\alpha^*{\bold G}$. If $\bold h \circ \tau$ is not of the form
$F(\bold k) - \bold k + \bold c$, where $\bold k \in W_l(K)$ and
$\bold c \in W_l(\F_q)$, then the result follows from Corollary 4.4
since the degree of the divisor $\bold G + \alpha^*\bold G$ is at most
$2\deg {\bold G}$.  

We now assume that $\bold h$ is of the type excluded by Corollary 4.4,
so that $\bold h = F(\bold k) - \bold k + \bold c$ for some $\bold k
\in W_l(K)$ and $\bold c \in W_l(\F_q)$.  
The first Witt coordinate of the equation $(\bold f - \bold g \circ
\alpha) \circ \tau = F(\bold k) - \bold k + \bold c$ is an equation of
the form $f_0 - g_0\circ\alpha = k_0^p-k_0 + c_0$. Now from our
hypothesis $f_0$ and $g_0$ have simple poles so $f_0 - g_0\circ\alpha$
also has simple poles. But $k_0^p-k_0 + c_0$ won't have simple poles
unless it is constant, so $f_0 - g_0\circ\alpha$ is constant.

If $\Delta \ne 0$, Lemma 5.3 ensures that $G$ and $\alpha^* G$ have
disjoint support and therefore $f_0$ and
$g_0\circ\alpha$ have disjoint polar divisors unless $f_0$ and
$g_0$ are both constants. Therefore, $f_0 - g_0\circ\alpha$ constant
implies that $f_0$ and $g_0$ are both constants. 
We take the (usual) Teichm\"uller lifts of
these constants and subtract them from ${\bold f}$, ${\bold
g}$, so that we may assume that $f_0, g_0$ are both zero.
This implies that $\bold f = p \bold f'$ and $\bold g = p \bold g'$
for some $\bold f', \bold g' \in \bold K$, where $\bold K$ is the
total quotient ring of rational functions on $\bold E$.
Using Lemma 2.4, we can write $\bold f = p \bold f'$, $\bold g = p
\bold g'$, where $\bold f'$ and $\bold g'$ are in the space of
global sections of the line bundle associated to ${\bold G}$.  As in
the proof of Theorem 3.1, we may consider $\bold f'$ and $\bold g'$ to
be defined over $W_{l-1}(K)$, so we use induction.  Either we have a
bound for the exponential sum formed with ${\bold f'} - {\bold
g'}\circ\alpha$ which is better than the bound in the statement of the
theorem, or we can repeat the process and finally get that ${\bold f}$
and ${\bold g}$ are constant, and this possibility is excluded by the
construction of our family.

If $\Delta = 0$, $\alpha$ is the identity, so $f_0 - g_0 = c_0$, a constant.
In order to compute correlations, the choice of the representative
$\bold f$ is immaterial, so we may subtract from $\bold f$ 
the (usual) Teichm\"uller lift of $c_0$ and assume that $c_0=0$.
As above ${\bold f} - {\bold g} = p {\bold h}$, for some ${\bold h}$
in $\Gamma({\bold E},\co_{\bold E}({\bold G}))$ and we can proceed by
induction on $l$ as before.

To estimate $|\cf|$, we note that $s_{\bold f}=s_{\bold g}$ implies that
the correlation of the two sequences is $n$, so by the above argument
${\bold f}-{\bold g}$ is a constant, hence 
$|\cf | = |\Gamma({\bold E},\co_{\bold E}({\bold G}))|/|W_l(\F_q)|$.
Finally, $\Gamma({\bold E},\co_{\bold E}({\bold G}))$ is a free 
$W_l(\F_q)$-module of rank $\deg {\bold G}$, by Theorem 2.2.
\enddemo





\bigskip


{\bf Acknowledgments:} The authors acknowledge the use of the software
packages Mathematica and Pari in some calculations.  The first author
would also like to thank A. Buium and J. Tate for useful comments and
the NSA (grant MDA904-97-1-0037) for financial support.  The second
author thanks Winnie Li and Patrick Sol\'e for their useful comments
and the NSF (grant DMS-9709388) for financial support.

\bigskip
\centerline{\bf References.}
\bigskip
\noindent
[Bu1] A.~Buium, {\it Geometry of $p$-jets}, Duke Math.~Journal {\bf 82}
(1996), 349--367.
\medskip
\noindent
[Bu2] A.~Buium, {\it An approximation property for Teichm\"uller points},
Math.~Research Letters, {\bf 3} (1996) 453--457.
\medskip
\noindent
[D] P.~Deligne: {\it Applications de la formule des traces aux sommes
trigonom\'etriques} in {\it Cohomologie \'etale (SGA 4$1 \over 2$)}, 
Lecture Notes in Math.~569, Springer-Verlag, Berlin, Heidelberg, 
New York (1977).
\medskip
\noindent
[G] V.~D.~Goppa, {\it Codes associated with divisors},
Probl.~Peredachi Inf. {\bf 13} (1977), 33-39.  (English translation in
{\it Probl.~Inf.~Transm.,} 13:22-27, (1977).) 
\medskip
\noindent
[HKCSS] A.~R.~Hammons, Jr., P.~V.~Kumar, A.~R.~Calderbank,
N.~J.~A.~Sloane, and P.~Sol\'e, {\it The $\Z_4$-Linearity of Kerdock,
Preparata, Goethals, and Related Codes}, IEEE Transactions on
Information Theory {\bf 40} (1994), 301-319.
\medskip
\noindent
[K] N.~M.~Katz, {\it Serre-Tate local moduli}, Springer LNM 868 (1981)  
138-202.
\medskip
\noindent
[KHC] P.~V.~Kumar, T.~Helleseth, and A.~R.~Calderbank,{\it An upper bound for 
some exponential sums over Galois rings and applications},
IEEE Transactions on Information Theory {\bf 41} (1995), 456-468.
\medskip
\noindent
[Li] W-C.~W.~Li, {\it Character sums over $p$-adic fields}, preprint (1997).
\medskip
\noindent
[LST] J.~Lubin, J-P.~Serre and J.~Tate, {\it Elliptic curves and formal  
groups},
Proc.~of the Woods Hole summer institute in algebraic geometry 1964.
Unpublished. Available at ¬http://www.ma.utexas.edu/users/voloch/lst.html.
\medskip
\noindent
[M] B. Mazur, {\it Frobenius and the Hodge filtration, estimates},
Ann. Math. {\bf 98} (1973) 58-95.
\medskip
\noindent
[S1] H.~L.~Schmid, {\it Zur Arithmetik der zyklischen $p$-K\"orper},
Crelles J., {\bf 176} (1936), 161-167.
\medskip
\noindent
[S2] H.~L.~Schmid, {\it Kongruenzzetafunktionen in zyklischen K\"orpern},
Abh.~Preuss. Akad.~Wiss.~Math.-Nat.~Kl. 1941, (1942). no.~14, 30 pp.
\medskip
\noindent
[St] H.~Stichtenoth, {\it Algebraic function fields and codes}, Springer,
Berlin, 1993.
\medskip
\noindent
[TV] M.~A.~Tsfasman and S.~G.~Vl\v adu\c t, {\it Algebraic-Geometric Codes},
Kluwer, Dordrecht, 1991.
\medskip
\noindent
[TVZ] M.~A.~Tsfasman, S.~G.~Vl\v adu\c t, and Th.~Zink, {\it Modular
curves, Shimura curves, and Goppa Codes, better than the
Varshamov-Gilbert bound}, Math.~Nachrichten, {\bf 109} (1982), 21-28.
\medskip
\noindent
[W1] J.~L.~Walker, {\it Algebraic geometric codes over rings},
submitted for publication.
\medskip
\noindent
[W2] J.~L.~Walker, {\it The Nordstrom Robinson code is algebraic
geometric}, IEEE Transactions on Information Theory {\bf 43} (1997),
1588-1593.
\medskip
\noindent
Dept.~of Mathematics, Univ.~of Texas, Austin, TX 78712, USA
\smallskip
\noindent
e-mail: voloch\@math.utexas.edu
\smallskip
\noindent
Dept.~of Mathematics and Statistics, Univ.of Nebraska, Lincoln, NE
68588-0323, USA
\smallskip
\noindent
e-mail: jwalker\@math.unl.edu



\end