Fibre products of Artin-Schreier curves and generalized Hamming weights of codes
Given an \([n, k, d]\)-code \(C\) its \(r\)-th generalized Hamming weight \(d_ r (C)\) is defined as the minimum of \(w(D)= \# S(D)\), where \(D\) runs through the \(r\)-dimensional subcodes of \(C\) and \(S(D)\) denotes its support. The \(r\)-dimensional weight distribution \(A^{(r)}_ i\) of \(C\) is the number of subcodes \(D\) with \(\dim D=r\) and \(w(D)=i\). The main purpose of the authors is to compute these numbers for certain trace codes obtained from fibre product of Artin-Schreier curves in terms of the number of rational points of these curves. More precisely, let \(P\) be a subset of \(\mathbb P^ 1 (\mathbb F_ q)\) and \({\mathcal L}\) a finite dimensional \(\mathbb F_ q\)-linear subspace of the field of rational functions \(\mathbb F_ q (x)\) with \({\mathcal L}\cap \mathbb F_ q= \{0\}\). Suppose that the non-zero elements of \({\mathcal L}\) have poles in \(P\) only and with multiplicities prime to \(p\), where \(q\) is a power of \(p\). Consider the trace code \[ \text{Tr}(C)= \left\{c_ f= \left(\text{Tr}(f(x))_{x\in \mathbb P^ 1 \left(\mathbb F_ q\right)-P}\right), \;f\in {\mathcal L}\right\}. \] For each \(r\)-dimensional \(\mathbb F_ q\)-subspace \(D\) of \(\text{Tr}(C)\), one chooses a basis \(c_{f_ 1}, \ldots, c_{f_ r}\) corresponding to \(f_ 1, \ldots, f_ r\) of \({\mathcal L}\) and denote by \({\mathcal L}_ D\) the subspace spanned by the \(f_ i\)'s. To each \(f_ i\) one attaches an Artin-Schreier curve \(C_{f_ i}\) with affine equation \(y^ p- y= f_ i (x)\) and considers the maps \(\varphi_ i: C_{f_ i}\to \mathbb P^ 1\). The authors compute the numbers \(d_ r (\text{Tr} (C))\) in terms of the number of rational points of the curve \(C^{(D)}\), which is the normalization of \(C_{f_ 1} \times_{\mathbb P^ 1} \cdots \times_{\mathbb P^ 1} C_{f_ i}\). This curve defines a family of curves \(C_ f\) in the Grassmann variety \(G(r,n)\) of \(r\)-planes in \(n\)-space, where \(n\) is the length of \(\text{Tr}(C)\). Further they estimate the difference \(d_ r (\text{Tr}(C))- (1-1/p^ r)n\) in terms of the genera of these curves. A similar result was independently obtained by \textit{H. Stichtenoth} and \textit{C. Voß} [IEEE Trans. Inf. Theory 40, No. 2, 554--558 (1994; Zbl 0837.94024)]. The authors apply this result to obtain new examples of supersingular curves of any genera (these are curves whose Jacobian is isogenous to a product of supersingular elliptic curves) using their former results [Compos. Math. 84, 333--367 (1992; Zbl 0804.14014)]. Moreover, they also obtain new examples of curves which achieve the maximum number of rational points according to the Hasse-Weil bound.
scientific article; zbMATH DE number 753624