Abstract
We study ${q}$ -ary linear codes ${C}$ obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of ${C}$ over all field extensions of $\mathbb {F}_{q}$ . Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code ${C}$ .
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