On List Decoding Transitive Codes From Random Errors

We study the error resilience of transitive linear codes over $\mathbb{F}_2$. We give tight bounds on the weight distribution of every such code $C$, and we show how these bounds can be used to infer bounds on the error rates that $C$ can tolerate on the binary symmetric channel. Using this connection, we show that every transitive code can be list-decoded from random errors. As an application, our results imply list-decoding bounds for Reed-Muller codes even when the rate exceeds the channel capacity.