van der Geer, Gerard; van der Vlugt, Marcel Trace codes and families of algebraic curves. (English) Zbl 0767.94018 Math. Z. 209, No. 2, 307-315 (1992). 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) Cited in 1 ReviewCited in 1 Document 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 Keywords:rational points; field; codes; subcodes; covering radius; weight distributions; curves; elliptic curves; integral binary quadratic forms PDFBibTeX XMLCite \textit{G. van der Geer} and \textit{M. van der Vlugt}, Math. Z. 209, No. 2, 307--315 (1992; Zbl 0767.94018) Full Text: DOI EuDML References: [1] [B] Bombieri, E.: On exponential sums in finite fields. Am. J. Math.88, 71–105 (1966) · Zbl 0171.41504 · doi:10.2307/2373048 [2] [CKMS] Cohen, G.C., Karpovsky, M.G., Mattson Jr., H.F., Schatz, J.R.: Covering radius-survey and recent results. IEEE Trans. on Inf. Theory31, 328–343 (1985) · Zbl 0586.94014 · doi:10.1109/TIT.1985.1057043 [3] [GV1] van der Geer, G., van der Vlugt, M.: Reed-Muller Codes and Super-singular Curves I. (Preprint 1990) · Zbl 0804.14014 [4] [GV2] van der Geer, G., van der Vlugt, M.: Kloosterman sums and thep-torsion of certain jacobians. Math. Ann.290, 549–563 (1991) · Zbl 0731.14014 · doi:10.1007/BF01459260 [5] [G] Goppa, V.D.: Codes associated with divisors. Probl. Peredachi. Inf.13, 33–39 (1977)=Probl. Inf. Transm.13, 22–27 (1977) · Zbl 0415.94005 [6] [HKM] Helleseth, T., Kløve, T., Mykkeltveit, J.: On the covering radius of binary codes. IEEE Trans. Inf. Theory24, 627–628 (1978) · Zbl 0379.94017 · doi:10.1109/TIT.1978.1055928 [7] [LW] Lachaud, G., Wolfmann, J.: Sommes de Kloosterman, courbes elliptiques et codes cycliques en caractéristique 2. C. R. Acad. Sci., Paris305, 881–883 (1987) · Zbl 0652.14009 [8] [SV] Schoof, R., van der Vlugt, M.: Hecke operators and the weight distribution of certain codes. J. Comb. Theory, Ser. A,57, 163–186 (1991) · Zbl 0729.11065 · doi:10.1016/0097-3165(91)90016-A [9] [W] Wolfmann, J.: New bounds on cyclic codes from algebraic curves In: Cohen, G., Wolfmann, J. (eds.) Coding Theory and Applications. (Lect. Notes in Comput. Sci., vol. 388) Berlin Heidelberg New York: Springer 1989 · Zbl 0678.94016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.