Yes, you most certainly can. You also have a risk of STD and STI transmission if you did not use a condom and other safer sex measures, which is increased for you both due to your period.
Being on one's period is not birth control. Neither is withdrawal. Neither are safer sex, either.
If it has been less then 72 hours since this occured, I suggest you get to a clinic and get emergency contraception. You should also make an appointment for STI screenings in about a month, as should your partner.
If it has been more than 72 hours, you will simply have to wait a month to take a pregnancy test, and them's the brakes.
I suggest in the interim that you are your partner look at the following together:
• Ready or Not? –The Readiness Checklist
• Safe, Sound & Sexy – A Safer Sex How-To
• Margaret Sanger’s Disneyland - An easy chart of your birth control options
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