First of all, the multiplicative depth $L$ and thus the selection of your ciphertext modulus $Q_L$ is important for guaranteeing a certain number ($L$) of correct ciphertext multiplications before needing to bootstrap the ciphertext. After once selecting $L$ resp. $Q_L$, it is fixed for the scheme, i.e. all future operations/encryptions. Apart from that, $L$ is an independent parameter, which has no "impact" on the correctness of the techniques of Encoding, Encrypting, Decrypting, etc.

Now turning to what I believe is what you wanted to know: Indeed there is an overflow of $a$, i.e. the uniformly sampled "random component" of the ciphertext. Especially when you multiply $a$ with the secret key and add an error and a message $m$, you should not be able to retrieve $m$, due to the fact that they are overflowing the modulus (by far) and thus look completely random afterward (RLWE security is based on that construction). But there is a technique to protect $m$ from small errors: Encoding $m$ via a scaling factor ($\Delta$ in the paper) ensures that everything gets decrypted correctly.