On the smallest size of an almost complete subset of a conic in \(\mathrm{PG}(2, q)\) and extendability of Reed-Solomon codes. (English. Russian original) Zbl 1411.94109
Probl. Inf. Transm. 54, No. 2, 101-115 (2018); translation from Probl. Peredachi Inf. 54, No. 2, 3-19 (2018).
Summary: Abstract In the projective plane \(\mathrm{PG}(2, q)\), a subset \(\mathcal{S}\) of a conic \(\mathcal{C}\) is said to be almost complete if it can be extended to a larger arc in \(\mathrm{PG}(2, q)\) only by the points of \(\mathcal{C}\setminus\mathcal{S}\) and by the nucleus of \(\mathcal{C}\) when \(q\) is even. We obtain new upper bounds on the smallest size \(t(q)\) of an almost complete subset of a conic, in particular,
\[t(q) < \sqrt{q(3\ln q + \ln\ln q + \ln 3)} + \sqrt{\frac{q}{{3\ln q}}} + 4 \sim \sqrt{3q\ln q},\]
\[t(q) < 1.835\sqrt{q\ln q.}\]
The new bounds are used to extend the set of pairs \((N, q)\) for which it is proved that every normal rational curve in the projective space \(\mathrm{PG}(N, q)\) is a complete \((q+1)\)-arc, or equivalently, that no \([q+1,N+1, q-N+1]_q\) generalized doubly-extended Reed-Solomon code can be extended to a \([q + 2,N + 1, q - N + 2]_q\) maximum distance separable code.
MSC:
| 94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |
| 51E21 | Blocking sets, ovals, \(k\)-arcs |
Software:
MagmaReferences:
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