In a breakthrough in the long-going attempt to construct good explicit tree codes, Cohen, Haeupler and Schulman (CHS) (STOC 2018) constructed constant-distance tree codes with rate 1/O(log log n). In their construction a large-alphabet tree code is used as a core element - and they were able to utilize polynomials over the Newton-basis to construct one, relying on a sparsity lemma for such polynomials (or alternatively, by using a construction of Pudlák). Here we simplify the proof of their result by replacing this element with an alternative construction, which consists of shifting input binary vectors and taking their binary sum. As an artifact of that, we note that the obtained rate-1/O(log log n) tree code is linear (a property which to our knowledge was not previously known).