The dual minimum distance of arbitrary-dimensional algebraic-geometric codes (Q420711)

If \(X\) is an algebraic variety \(X\) defined over a finite field of dimension \(>1\), then the dual of an algebraic-geometric valuation code associated to \(X\) (and a divisor on \(X\) and a set of points whose support is disjoint form that divisor) need not be a algebraic-geometric valuation code associated to \(X\). The question arisies then, is \textit{dual} of an algebraic-geometric valuation code associated to \(X\) a code with ``good'' parameters? This motivates the paper under review.NEWLINENEWLINEIn this paper, the author studies the minimum distance of the dual \(C^\perp\) of an algebraic-geometric valuation code associated to an algebraic variety \(X\) defined over a finite field of dimension \(\geq1\). The approach consists in describing the minimal configurations of points on \(X\) which fail to impose independent conditions on forms of some degree. In some cases, the result improves the well-known Goppa designed distance when \(X\) is a curve (see [\textit{M. Tadic}, Glas. Mat., III. Ser. 37, No. 1, 21--57 (2002; Zbl 1007.22021)]).