Its basic principle is to use the relationship between the roots and coefficients of higher-order equations in a variable. We can solve one-dimensional linear equation, one-dimensional quadratic equation and special one-dimensional cubic equation, and we can't go any higher.
In this respect, there is a person worth mentioning, that is Galois, who found a method to solve the unary equation of higher order. That is, each equation corresponds to a domain, that is, the domain containing all the roots of the equation, which is called the Galois field of this equation, and this domain corresponds to a group, that is, the permutation group of the roots of this equation, which is called the Galois group of this equation. There is a one-to-one correspondence between the subdomains of Galois field and the subgroups of Galois group. An equation is completely solvable if and only if its Galois group is solvable. If you are really interested, you can find some related works to study. I'm from Soft Power (Beijing) Co., Ltd., and I've been doing QR codes for almost ten years. The error correction codec we use is modified and tested on the basis of predecessors, and there is not much research on the error correction algorithm itself.
I hope the above can provide you with some clues and basic help, and I hope you can become the pride of the Chinese nation in mathematics.