Bill,
I sent Ruger Customer Service an email asking them if my serial number was a "heavy barrel" version. This is the text of the email they sent me back:
Thank you for contacting Ruger Customer Service.
Your Customer Service Issue # is 10095189
Yes it is a SS-34H. 158-064XX is a Speed-Six revolver .357 magnum, model number 01543, blued, 4" barrel, walnut round butt grip panels, fixed sights, shipped in November of 1981.
Ruger Customer Service
The "H" in SS-34H does stand for heavy barrel, right?
If you send me a PM with your email address I'd be happy to forward you the email they sent me.
Robert