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Weights in codes and genus 2 curves. (English) Zbl 1077.94029

Let \(q=2^m\), with \(m>2\), and let \(\alpha\) be a primitive element in \({\mathbb F}_q\). Let \(m_i(x)\) denote the minimal polynomial of \(\alpha^i\) over \({\mathbb F}_2\). The authors apply geometric methods to obtain information about the binary cyclic code \(C\) of length \(q-1\) with generator polynomial \(m_{-1}(x)m_1(x)m_3(x)\). Since \(C\) is a subcode of the 2-error-correcting BCH code, the minimum distance of \(C\) is at least 5. The authors show that a codeword of weight 5 exists for all \(m\geq 16\) by showing that a certain space curve, defined as the intersection of two surfaces, must have more than four \({\mathbb F}_q\)-rational points. Next, they consider the weights of codewords in \(C^\perp\). By the usual technique of applying Delsarte’s Theorem and Hilbert’s Theorem 90, the weights of codewords in \(C^\perp\) are related to the number of rational points on curves of the form \(y^2+y=ax^{-1}+bx+cx^3\). To see the possibilities for the number of such rational points, the authors use results due to H.-G. Rück [Compos. Math. 76, No. 3, 351–366 (1990; Zbl 0742.14037)] and D. Maisner and E. Nart [Exp. Math. 11, No. 3, 321–337 (2002)] on the possible zeta functions of such genus two curves, as well as a construction due to G. Frey and E. Kani [Arithmetic algebraic geometry, Proc. Conf., Texel/Neth. 1989, Prog. Math. 89, 153–176 (1991; Zbl 0757.14015)]. For \(m\) even, they give a precise description of which weights occur in \(C^\perp\). For \(m\) odd, they specify an interval \(I\) that contains all the weights, and a subinterval \(J\) such that every even integer in \(J\) occurs as a weight, but they cannot characterize which even integers in \(I\setminus J\) occur as weights. The problem of determining the distribution of the weights remains open.

MSC:

94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
94B15 Cyclic codes
11G10 Abelian varieties of dimension \(> 1\)
11G20 Curves over finite and local fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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References:

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