Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. (English) Zbl 1348.11052
The authors build on the work of A. Sboui by studying the weights of the \(d\)-th order \(q\)-ary projective Reed-Muller codes \(\mathrm{PRM}(q,d,n)\), for small \(d\). This is accomplished by finding bounds on the number of points in algebraic curves and hypersurfaces in \(P^n(F_q)\), depending on the number of linear components contained in these objects.
Reviewer: Steven T. Dougherty (Scranton)
MSC:
| 11G25 | Varieties over finite and local fields |
| 14G50 | Applications to coding theory and cryptography of arithmetic geometry |
| 94B05 | Linear codes (general theory) |
Keywords:
algebraic variety; quadrics; small weight codewords; intersections; projective Reed-Muller codesReferences:
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