When we construct one-point algebraic geometry codes, we have to find a basis of

LetMore detailed statements and their complete proofs are described in AMS LaTeX 1.2 format, DVI format, PDF format, and PostScript format.Vbe an plane algebraic set defined by a bivariate polynomial of form \[ c_{b,0} X^b + c_{0,a} Y^a + \sum_{ai + bj < ab} c_{i,j} X^i Y^j. \] ThenVis an algebraic curve with a unique rational placeQ, and pole divisors ofXandYareaQandbQrespectively. IfVis nonsingular, then a basis ofL(mQ)is \[ \{ X^i Y^j | 0 \leq i, 0 \leq j \leq a-1, ai + bj \leq m \}. \] Elements in the basis have pairwise distinct discrete valuations atQ.

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