On the generalized weights of a class of trace codes (Q5941627)

The authors define a class of codes by considering as a coordinate set the set of points \(X\) where the trace of a hermitian form \(f\) evaluated at a point is zero \((X = \{x|\text{Tr}(f(x)) = 0\}\)). The codewords consist of traces of linear forms evaluated at the coordinate set \(X(\text{Tr}(l(x))_{x\in X})\). By using results on exponential sums, the \(r\)-th mimimum distance (\(r\)-th generalized Hamming weight) can be expressed in the number of zeros of certain polynomials connected to the linear forms. This leads in a trivial way to lower (and also to upper) bounds for the \(r\)-th minimum distance. Next the authors show that under certain conditions these lower bounds can be attained by explicitly giving subspaces for which equality holds. If one is interested in the results of this paper it is advisable to read the authors' reference [2; see Zbl 0856.11057] first. Unfortunately this reference as given contains an error: one can find it on pages 1636-1642 of IEEE Trans. Inf. Theory 41, No.~6, November 1995 , and not on pages 1-6 of No. 5 (1995) as stated in the paper. The authors do not give any clue to how their results compare to other known results.