I already read the link a few hours before it was posted. There was zero details on the algorithm and no link to the actual research that I could see.
Well, in case you are interested, after checking around, it appears that this "algorithm" was a minor result of Mr. Helfgott's work to prove the ternary Goldbach conjecture (every odd integer n greater than 5 is the sum of three primes). Here's the preprint of the paper, I should warn you that it appears to be a very theoretical paper, one targetted at the Goldbach conjecture (not practical prime sieving), so there is not a fully fleshed out algorithm that you can translate into a computer program. I haven't gone through the paper in detail, but it appears to rely heavily on technique from a Messr. Ramare.
We start by adapting ideas from Ramare’s version of the large sieve for primes to estimate l2 norms over parts of the circle. We are left with the task
of giving an explicit bound on the factor in Ramare’s work. As a side effect, this finally gives a fully explicit large sieve for primes that is asymptotically optimal, meaning a sieve that does not have a spurious factor of exp(gamma) in front; this was an arguably important gap in the literature.
I cannot find a definitive paper about this technique, but is appears to be related to this earlier paper. Mr. Helfgott apparently just tightening the bounds which theoretically should create a better sieve algorithm. My impression is that I think it will take some concerted effort to create a computer algorithm out of this algorithm.
However, your mileage may vary...