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Maximum distance separable codes and arcs in projective spaces. (English) Zbl 1126.94017

An $\left(n,k,q\right)$-MDS code $C$ is a collection of ${q}^{k}$ distinct $n$-tuples, called codewords, over an alphabet $𝒜$ of size $q$, satisfying the following condition: no two codewords of $C$ agree in as many as $k$ coordinate positions. The importance of these MDS codes is that they satisfy the Singleton bound of coding theory, that is, they are codes of length $n$, containing ${q}^{k}$ codewords, and whose minimal distance $d$ is equal to $d=n-k+1$.

Linear $\left[n,k,n-k+1\right]$-MDS codes over the finite field of order $q$ are equivalent to $n$-arcs in PG$\left(k-1,q\right)$ and to $n$-arcs in PG$\left(n-k-1,q\right)$. This geometrical link to arcs has made it possible to prove many results on linear MDS codes. In particular, great attention has been paid to the problem of the extendability of $n$-arcs in PG$\left(k-1,q\right)$ to $\left(n+1\right)$-arcs in PG$\left(k-1,q\right)$; in this way studying the problem of the extendability of the corresponding $\left[n,k,n-k+1\right]$-MDS codes to $\left[n+1,k,n-k+2\right]$-MDS codes.

In some cases, the non-extendability of linear $\left[n,k,n-k+1\right]$-MDS codes to linear $\left[n+1,k,n-k+2\right]$-MDS codes is known. But could these codes be extended to non-linear MDS codes of length $n+1$?

The authors contribute to this particular extendability problem. They obtain new results by using new geometrical links. The new links are with Rédei-type blocking sets.

Consider an affine plane $A$ of order $q$, with line $\ell$ at infinity. Let $\pi$ be the projective plane defined by $A$ and $\ell$. A Rédei-type blocking set of $\pi$, w.r.t. the line $\ell$, is a set $B$ consisting of $q$ points of $A$, together with the intersection points of all secants to $A$ with the line $\ell$.

These Rédei-type blocking sets in PG$\left(2,q\right)$ have been studied in great detail by A. Blokhuis, S. Ball, A.E. Brouwer, L. Storme and T. Szőnyi [J. Comb. Theory, Ser. A 86, No. 1, 187–196 (1999; Zbl 0945.51002)] and S. Ball [J. Comb. Theory, Ser. A 104, No. 2, 341–350 (2003; Zbl 1045.51004)]. Let ${𝒫}_{q}>1$ be the smallest size for the intersection $B\cap \ell$ of a Rédei-type blocking set $B$, different from a line, w.r.t. $\ell$.

To illustrate the link between the extendability problem of linear MDS codes and Rédei-type blocking sets of PG$\left(2,q\right)$, we mention the following result: Let $C$ be a linear $\left[n,3,n-2\right]$-MDS code, with $n>q+2-{𝒫}_{q}$. Then any arbitrary extension of $C$ to an MDS code of length $n+1$ must be linear.

We also wish to mention with respect to Section 7.1 the results of L. Storme and J. A. Thas [J. Comb. Theory, Ser. A 62, No. 1, 139–154 (1993; Zbl 0771.51006)] on arcs in PG$\left(N,q\right)$, $q$ even.

##### MSC:
 94B25 Combinatorial codes 05B25 Finite geometries (combinatorics) 51E14 Finite partial geometries (general), nets, partial spreads 51E20 Combinatorial structures in finite projective spaces 51E21 Blocking sets, ovals, $k$-arcs
##### Keywords:
MDS codes; code extensions; linear codes; arcs; dual arcs; complete arcs