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Trace codes and families of algebraic curves. (English) Zbl 0767.94018

Let \(p\) be a prime and let \(\mathbb F_ q\) be a field with \(q=p^ a\) elements. Consider the following codes over \(\mathbb F_ p\): \[ C_ R(q)=\left\{c(a,b)=\left(\text{Tr}\left(ax+{b\over R(x)}\right)\right)_{x\in\mathbb F_ q-Z}: a,b\in\mathbb F_ q\right\}. \] Here \(R(x)=\sum_ i a_ ix^{p^ i}\in\mathbb F_ q[x]\) and \(Z\) denotes the \(\mathbb F_ p\)-vector space of zeroes of \(R\) in \(\mathbb F_ q\). By Tr we denote the trace map from \(\mathbb F_ q\) to \(\mathbb F_ p\).
The authors show that the subcodes \(\{c(a,b): a,b\in\mathbb F_ q\) for which \(\text{Tr}(ax)=0\) for all \(x\in Z\}\) have covering radius \(\left(1- {1\over p}\right)\bigl(q-p^{\dim(Z)}\bigr)\). In the special case \(p=2\) and \(R(x)=x^ 2+x\) the authors determine the weight distributions of the codes \(C_ R(q)\). Briefly, the weights are related to the number of \(\mathbb F_ q\)-rational points on the curves in the family. \(Y^ 2+Y=aX+b/X(X+1)\), \(a,b\in\mathbb F_ q\). For \(a,b\in\mathbb F_ q^*\), this curve has genus 2 and its Jacobian is isogenous to a product of two elliptic curves. The final answer involves the number of elliptic curves over \(\mathbb F_ q\) with a given number of rational points and is given in terms of certain products of class numbers of integral binary quadratic forms.
Reviewer: R.Schoof (Povo)

MSC:

94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
14G50 Applications to coding theory and cryptography of arithmetic geometry
14G15 Finite ground fields in algebraic geometry
14G05 Rational points
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References:

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