Linear recursive MDS-codes of dimensions~2 and~3

A. S. Abashin

Diskretnaya Matematika, Tome 12 (2000) no. 2, pp. 140-153

Résumé

A code $\mathcal K$ of length $n$ in an alphabet $\Omega$ is called linear in the general sense or simply linear if there exists a binary operation $+$ on $\Omega$ such that $(\Omega,+)$ is an abelian group and $\mathcal K$ is a subgroup of $(\Omega^n,+)$. We say that $\mathcal K$ is a $k$-recursive code if $\mathcal K$ consists of all words of length $n\ge k$ such that their coordinates are obtained from the first $k$ coordinates by some fixed recursive rule. Let $l^r(k,q)$ be the maximal $n$ such that there exists a linear $k$-recursive code of length $n$ in an alphabet of $q$ elements with the distance $n-k+1$ (an MDS code), and let $l^{ir}(k,q)$ be the maximal $n$ such that there exists a linear $k$-recursive idempotent (containing all constant words) MDS code of length $n$ in an alphabet of $q$ elements. Using the theory of linear recurring sequences we find $l^{ir}(2,q)$ and $l^{r}(3,q)$ for primary $q$.