Maximum distance separable codes to order

Abstract

Maximum distance separable (MDS) are constructed to required specifications. The codes are explicitly given over finite fields with efficient encoding and decoding algorithms. Series of such codes over finite fields with ratio of distance to length approaching $(1-R)$ for given $R, \, 0 < R < 1$ are derived. For given rate $R=\frac{r}{n}$, with $p$ not dividing $n$, series of codes over finite fields of characteristic $p$ are constructed such that the ratio of the distance to the length approaches $(1-R)$. For a given field $GF(q)$ MDS codes of the form $(q-1,r)$ are constructed for any $r$. The codes are encompassing, easy to construct with efficient encoding and decoding algorithms of complexity $\max\{O(n\log n), t^2\}$, where $t$ is the error-correcting capability of the code.