TITLE: Coding Theory and Algebra
QUESTION [8 upvotes]: Why is Algebra so useful in Coding Theory?
I know a little bit of algebra and I just know what codes are. I appreciate it if someone can give a brief explanation of how/ in what sense is algebra useful in Coding Theory.
More specifically, I recently asked a question about Hensel's Lemma and it's use and got Coding Theory as answer. I read online that factorization of polynomials has something to do with it, but still am unable to firmly grasp the relation of Coding and Hensel's Lemma.
I appreciate some explanations or reference to any "good" source.

REPLY [4 votes]: As an addition to rschwieb's answer: Good codes often come with a lot of symmetries, which gives a connection to group theory. For example, the automorphism group of the binary extended Golay code is the Mathieu group $M_{24}$, which is one of the sporadic simple groups.
As textbooks on algebraic coding theory I recommend:

"The theory of error-correcting codes" by MacWilliams and Sloane. Written in the 1970s and mostly still state of the art, it is the classical book on coding theory.
For newer developments (already mentioned by rschwieb): "Fundamentals of error-correcting codes" by Huffman and Pless. Its chapter 12.4 is on cyclic codes over the ring $\mathbb Z/4$ and contains the application of Hensel's lemma mentioned in the question.