  This paper gives new methods of constructing {\it symmetric self-dual codes}
over a finite field $GF(q)$ where $q$ is a power of an odd prime. These methods
are motivated by the well-known Pless symmetry codes and quadratic double
circulant codes. Using these methods, we construct an amount of symmetric
self-dual codes over $GF(11)$, $GF(19)$, and $GF(23)$ of every length less than
42. We also find 153 {\it new} self-dual codes up to equivalence: they are
$[32, 16, 12]$, $[36, 18, 13]$, and $[40, 20,14]$ codes over $GF(11)$, $[36,
18, 14]$ and $[40, 20, 15]$ codes over $GF(19)$, and $[32, 16, 12]$, $[36, 18,
14]$, and $[40, 20, 15]$ codes over $GF(23)$. They all have new parameters with
respect to self-dual codes. Consequently, we improve bounds on the highest
minimum distance of self-dual codes, which have not been significantly updated
for almost two decades.
