An error-correcting code is a set containing codewords, i.e., sequences of symbols from a specified finite alphabet, such that even after a few symbols of a codeword are altered it is possible to recover the codeword. Error-correcting codes lie at the heart of the digital era we live in, and are needed to protect digital data from constant interference from nature and adversaries.

From the very beginnings of the field in the early 1950's it became clear that algebra over finite fields leads to amazing constructions of error-correcting codes, thus addressing fundamental combinatorial challenges. What is less obvious is that more sophisticated (but still often quite elementary) algebra also leads to efficient algorithms that find and eliminate the errors in information transmission --- solving tasks in polynomial time that on the surface seem to require brute force search. In this talk we will survey some of the simple, and more sophisticated, ideas that lead to these effective algorithms and the limits they achieve.