# Error Check in Lattice PQC

I am by no means an expert in the PQC field and am just trying to self teach myself about it. I was hoping to look into error correcting in lattice. I want learn about error or fault detection as it applies to lattice, but I am unable to find many resources.

Does something like CRC apply to a lattice? Or are there lattice specific mechanisms?

I know this is a little broad, just looking for some insights.

2. Error-correction with "lattice codes", or more properly codes for the Additive White Gaussian Noise (AWGN) channel. These are often called "sphere packings", and are used to correct the LWE error $$e$$ in an LWE sample $$(A, As + e)$$, which is of small $$\ell_p$$ norm (generally for $$p = \infty$$).
Lattice cryptography very often uses this second type of code. Basic examples are in Regev-style encryption, where one encrypts $$m\in\{0,1\}$$ via $$(A, As + e + (q/2)m)$$. Here, one can view $$(q/2)m$$ as the value of $$m$$ being encoded under the lattice code corresponding to $$(q/2)\mathbb{Z}^n$$. There are other more complex lattice codes used --- the Micciancio-Piekert "gadget matrix" $$G$$ can be seen as using the lattice $$\bigoplus_i\Lambda_q(g^t)$$, where $$g = (1,2,\dots,2^{k-1})$$. Van Poppelen's masters thesis looked into the potential benefits of using (direct sums of) the Leech lattice $$\Lambda_{24}$$.