  The mds (maximum distance separable) conjecture claims that a nontrivial
linear mds $[n,k]$ code over the finite field $GF(q)$ satisfies $n \leq (q +
1)$, except when $q$ is even and $k = 3$ or $k = q- 1$ in which case it
satisfies $n \leq (q + 2)$.
  For given field $GF(q)$ and any given $k$, series of mds $[q+1,k]$ codes are
constructed.
  Any $[n,3]$ mds or $[n,n-3]$ mds code over $GF(q)$ must satisfy $n\leq (q+1)$
for $q$ odd and $n\leq (q+2)$ for $q$ even. For even $q$, mds $[q+2,3]$ and mds
$[q+2, q-1]$ codes are constructed over $GF(q)$.
  The codes constructed have efficient encoding and decoding algorithms.
