The module is AM and the noise is quite normal. A logic 1 means the transmitter is turned on, a logic zero turns it off (nothing transmitted at all). When the transmitter is off, the receiver picks up anything on the frequency, including it's own internal noise. You can't stop it but you can protect against it in several ways:
1. Use a monostable at the receiver output. Set it to the bit length of the data you are using, that will help to eliminate anything shorter than one bit.
2. If your data is mostly zero bits, try inverting it at the transmitter and receiver. The noise is produced when the receiver is not picking up your transmission and hears background inteference instead. If inverting the bits makes the transmission last longer there will be less chance of interference being picked up.
3. Try turning the transmitter on for a short period (~100mS) before sending data to allow the receiver to adapt to the signal level.
The other alternative is to encode the data in such a way that it is less susceptible to corruption but this method will involve software changes at the transmitter and receiver. Personally, I gave up on AM transmitters a long time ago. FM types are almost the same price and far more robust.
Brian.