Codes coming from a blowing up of the plane
Let \(\mathbb{F}_q\) be a finite field with \(q\) elements and let \(V=V(xyz)\subset \mathbb{P}^2(\mathbb{F}_q)\) be the union of the three axes of the projective plane. Fix a family of points \(P_1,\dots,P_b\in V\) with \(b\geq 2\) and a family of integers \(m_1,\dots,m_b\geq2\). Suppose that \(q\geq m+2\geq m_1+\cdots+m_b+1\) for a certain integer \(m\). Then, \textit{J. A. Davis} considered in [J. Pure Appl. Algebra 215, No. 4, 496--510 (2011; Zbl 1205.14029)] the \([n,k,d]\)-code obtained by evaluating at all points of \(\mathbb{P}^2(\mathbb{F}_q)\setminus V\) all homogeneous polynomials of degree \(m\) vanishing at each \(P_i\) with multiplicity at least \(m_i\). This code has \(n=(q-1)^2\) and \[ k=\begin{pmatrix} m+2\\2\end{pmatrix}-\sum\nolimits_{1\leq i\leq b}\begin{pmatrix} m_i+1\\2\end{pmatrix}. \] For the minimum distance \(d\) a lower bound was given. In the paper under review, the lower bound for \(d\) is slightly improved in the case \(q\geq m+3\).
scientific article; zbMATH DE number 6150680