It is tempting to say that paradigm-shift coming with turbo and LDPC codes instead of earlier popular approaches: convolutional codes, Reed-Solomon codes, BCH codes et.al. is shift from algebra to probability, from order to chaos. I mean that earlier constructions were much dominated by algebra considerations e.g. non-recursive convolutional codes are just the ideals in the ring $F_2[x]\oplus ... \oplus F_2[x]$. While turbo and LDPC are actually constructed and decoded with methods which much influenced by probabilistic and randomized considerations: roughly speaking good LDPC codes can be constructed by sufficiently sparse and random matrix. The decoding methods used for LDPC - belief propagation naturally belong to probability or machine learning maths. rather than algebra.
Actually turbo code is almost the same as convolutional code, modula one "small" detail - interleaver. Interleaver is "radomizer" added to the algebra-tasted convolutional code, it is crucial thing which makes all work. That what concerns the encoder. The decoder of turbo-codes "resembles" turbine and hence the name "turbo"-code, it is crucially based on probabilistic techniques in coding theory. Well, the key technique - BCJR algorithm was developed much earlier, so, of course, all division into old-new paradigms is not very precise, but nevertheless seems there is something behind it.
These ideas found rich practical applications. If someone is reading this with the help of smartphone - say thank to "turbo-codes" - they are working there.
New discovery - polar codes - probably can be characterized as algebra's strike back - they seems to be quite algebraic nature, sorry I cannot say much for the moment.