MDS Constacyclic Codes and MDS Symbol-Pair Constacyclic Codes

Symbol-pair codes are used to protect against symbol-pair errors in high density data storage systems. One of the most important tasks in symbol-pair coding theory is to design MDS codes. MDS symbol-pair codes are optimal in the sense that such codes attain the Singleton bound. In this paper, a new class of MDS symbol-pair codes with code-length
$5p$
and optimal pair distance of 7 is established. It is shown that for any prime
$p \equiv 1 \pmod 5$
, we can always construct four
$p$
-ary MDS symbol-pair cyclic codes of length
$5p$
of largest possible pair distance 7. We also completely determined all MDS symbol-pair and MDS
$b$
-symbol codes of length
$p^{s}$
and
$2p^{s}$
over
$\mathbb F_{p^{m}}+u\mathbb F_{p^{m}}$
by filling in some missing cases, and rectifying some gaps in Type 3 codes of recent papers. As an applications of our results, we use MAGMA to provide many examples of new MDS codes over
$\mathbb F_{p^{m}}$
and
$\mathbb F_{p^{m}}+u\mathbb F_{p^{m}}$
.