Self-Dual Negacyclic Codes With Variable Lengths and Square-Root-Like Lower Bounds on the Minimum Distances

The construction of self-dual codes with large minimum distances has been an active topic in coding theory. The construction and classification of extremal self-dual codes over small fields are related to other fields of mathematics, such as lattices and invariant theory as well as combinatorial
$t$
-designs. It is well-known that
$q$
-ary self-dual cyclic codes exist only when
$q$
is an even prime power and
$q$
-ary self-dual negacyclic codes exist for any odd prime power
$q$
. In 2009 a family of binary self-dual cyclic codes with lengths
$n_{i}$
and minimum distances
$d_{i} \geq \frac {1}{2} \sqrt {n_{i}}$
, where
$n_{i}$
goes to the infinity if
$i$
goes to the infinity, was constructed. In this paper, we construct several families of
$q$
-ary self-dual negacyclic codes of lengths
$n$
with their minimum distances larger than or equal to
$n^{\frac {1}{2}}$
for various lengths
$n$
and any given odd prime power
$q$
. When
$q \in \{3,5\}$
and the length is small, the minimum distances of the constructed self-dual negacyclic codes are comparable with these self-dual codes with largest known minimum distances in the literature.