On M.D.S. codes, arcs in PG(n,q) with q even, and a solution of three fundamental problems of B. Segre
Let C be a code of length n over an alphabet A of size q, \(q\geq 2\), \(| A| =q\). If no two words agree in k positions and \(| C| =q\) k then C is called a maximum distance separable (MDS) code. A question of interest is, given k,q what is the maximum value of n for which such codes exist? In the case when C is linear over \(F=GF(q)\) it can be generated by a \(k\times n\) matrix X over F of rank k, in which every set of k columns are independent. The columns of X are regarded as points in a projective space of dimension k-1, PG(k-1,q). The columns of X then form a k-arc in the space. Thus questions of s-arcs in the space relate directly to MDS codes. The following questions were posed by \textit{B. Segre} [Ann. Mat. pura appl., IV. Ser. 39, 357-379 (1955; Zbl 0066.140)]: a) For given r and q what is the maximum value of s for which there exist s-arcs in PG(r,q). b) For what values of r and q, with \(q>r+1\), is every \((q+1)\)-arc of PG(r,q) the point set of a normal rational curve? c) For given r, and q, with \(q>r+1\), what are the values of s for which every s-arc of PG(r,q) is contained in a normal rational curve of this space? Substantial progress is reported here on these three problems with attention confined to the case of q even.
scientific article; zbMATH DE number 4068780
scientific article