Algebraic codes and Invariance Madhu Sudan (Harvard) Algebraic error-correcting codes hold a central place in coding theory due to three, potentially unrelated, features. The most well-known feature is perhaps the combinatorial aspect: (1) Algebraic code pack Hamming space extremely densely, often outperform random error-correcting codes. Less well-known are the (2) Multiplicative property: Algebraic codes come endowed with a product operation where products of codewords remain far from each other and (3) Symmetries: Many algebraic codes are endowed with a rich group of symmetries that enable powerful uses of these codes. In this talk I will briefly review algebraic codes and the first two properties, before turning to the third part and talk about recent works highlighting the role of symmetries in (mathematical) uses of error-correcting codes.