The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces

Frédéric A. B. Edoukou

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Differential approach for the study of duals of algebraic-geometric codes on surfaces

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The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code $C_{L} (D, G)$ on a curve is the differential code $C_{Ω} (D, G)$ . We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study...

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Let C be a smooth non-degenerate integral curve of degree d and genus g in $ℙ^{3}$ over an algebraically closed field of characteristic zero. For each point P in $ℙ^{3}$ let $V_{P}$ be the linear system on C induced by the hyperplanes through P. By $V_{P}$ one maps C onto a plane curve $C_{P}$ , such a map can be seen as a projection of C from P. If P is not the vertex of a cone of bisecant lines, then $C_{P}$ will have only finitely many singular points; or to put it slightly different: The secant scheme $S_{P} = {(V_{P})}_{2}^{1}$ parametrizing...

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Minimal Codewords in Linear Codes

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2000 Mathematics Subject Classification: 94B05, 94B15. Cyclic binary codes C of block length n = 2^m − 1 and generator polynomial g(x) = m1(x)m2^s+1(x), (s, m) = 1, are considered. The cardinalities of the sets of minimal codewords of weights 10 and 11 in codes C and of weight 12 in their extended codes ^C are determined. The weight distributions of minimal codewords in the binary Reed-Muller codes RM (3, 6) and RM (3, 7) are determined. The applied method enables codes...