Impossible to reinstall Piwik

I have a blog analyzed by Piwik.
My Batabase MySql on the server was overall. So, I decided to clean all Piwik information by deleting all Piwik tables on the server with phpMyAdmin and by removing all Piwik files from the FTP blog.
I don’t know too much about PHP, Piwik… but I was thinking that I have removed everything about Piwik.

After, I reinstall Piwik (1.8.2). Piwik give on the first page the following error message:
SQLSTATE[42S02]: Base table or view not found: 1146 Table ‘xxxxxxblog.piwik_option’ doesn’t exist

I don’t understand how Piwik succeeds to find information from the last installation. I was thinking that I cleaned it totally.
Can someone help me to reinstall Piwik?

Thank you for any help.

delete config/config.ini.php and reinstall piwik with a different DB Prefix maybe ?

Thank you Matt.
I do not understand how the file config/config.ini.php succeeded to be in place after I removed the full repertory Piwik by FTP…
Now Piwik is launched.
Thank you for your help.