The following sorting network for 29 inputs has 165 compare-exchange-units, which is one less that the most size-optimal 29-input sorting networks that I could find in the litterature. Here is how I generated it: first it sorts the first 16 inputs and the last 13 inputs independently. Then it merges the two sorted subarrays using a size 32 Batcher odd-even merge network (the version that does not need the inputs to be interleaved), where all compare-exchange units working on indexes greater than 28 have been dropped. Dropping comparators in such a way is ok: consider that the values at the indexes [29, 32) are greater than every other value in the array to sort, and it will become intuitive that dropping them generates a correct merging network of a smaller size.
That said, even though I have been unable to find a 29-input sorting network with as few compare-exchange units as 165 in the litterature, I can't claim that I found the technique used to generate it: the unclassified 1971 paper [A Generalization of the Divide-Sort-Merge Strategy for Sorting Networks][8] by David C. Van Voorhis already describes the as follows:
The improved 26-,27-,28-, and 34-sorters all use two initial sort units, one of them the particularly efficient 16-sorter designed by M. W. Green, followed by Batcher's [2,2] merge network.
The paper does not mention a better result than 166 CEUs for the 29-input sorting networks, but that's only because our solution relies on a 13-input sorting networks that uses 45 CEUs, while the best known such network in 1971 used 46 CEUs. I couldn't find any resource using the technique to improve the 29-input sorting network since then, even though some of them mention a 156-CEU 28-input sorting network that has apparently only been described in the aforementioned unclassified paper.
[[0, 1],[2, 3],[4, 5],[6, 7],[8, 9],[10, 11],[12, 13],[14, 15],[17, 23],[25, 27]]
[[0, 2],[4, 6],[8, 10],[12, 14],[19, 20],[21, 24]]
[[1, 3],[5, 7],[9, 11],[13, 15],[16, 28]]
[[0, 4],[8, 12],[18, 22]]
[[1, 5],[9, 13],[16, 17],[18, 19],[20, 22],[24, 27]]
[[2, 6],[10, 14],[23, 28]]
[[3, 7],[11, 15],[21, 25]]
[[0, 8],[16, 18],[19, 23],[26, 27]]
[[1, 9],[17, 20],[22, 28]]
[[2, 10],[23, 24],[27, 28]]
[[3, 11],[20, 25]]
[[4, 12],[22, 26]]
[[5, 13],[19, 20],[21, 22],[24, 25],[26, 27]]
[[6, 14],[17, 23]]
[[7, 15],[18, 22],[25, 27]]
[[5, 10],[17, 19],[20, 23],[24, 26]]
[[6, 9],[16, 21]]
[[3, 12],[13, 14],[1, 2],[18, 21],[22, 24],[25, 26]]
[[7, 11],[17, 18],[19, 21],[23, 24]]
[[4, 8],[20, 22]]
[[1, 4],[7, 13],[18, 19],[20, 21],[22, 23],[24, 25]]
[[2, 8],[11, 14],[19, 20],[21, 22]]
[[2, 4],[5, 6],[9, 10],[11, 13]]
[[3, 8]]
[[7, 12]]
[[6, 8],[10, 12]]
[[3, 5],[7, 9]]
[[3, 4],[5, 6],[7, 8],[9, 10],[11, 12]]
[[6, 7],[8, 9]]
[[0, 16]]
[[8, 24]]
[[8, 16]]
[[4, 20]]
[[12, 28]]
[[12, 20]]
[[4, 8],[12, 16],[20, 24]]
[[2, 18]]
[[10, 26]]
[[10, 18]]
[[6, 22]]
[[14, 22]]
[[6, 10],[14, 18],[22, 26]]
[[2, 4],[6, 8],[10, 12],[14, 16],[18, 20],[22, 24],[26, 28]]
[[1, 17]]
[[9, 25]]
[[9, 17]]
[[5, 21]]
[[13, 21]]
[[5, 9],[13, 17],[21, 25]]
[[3, 19]]
[[11, 27]]
[[11, 19]]
[[7, 23]]
[[15, 23]]
[[7, 11],[15, 19],[23, 27]]
[[3, 5],[7, 9],[11, 13],[15, 17],[19, 21],[23, 25]]
[[1, 2],[3, 4],[5, 6],[7, 8],[9, 10],[11, 12],[13, 14],[15, 16],[17, 18],[19, 20],[21, 22],[23, 24],[25, 26],[27, 28]]
I verified this network using all 229 binary test vectors, which took about 5 minutes. My test app says the depth is 31. If you don't plan to publish this formally, consider sending this solution to Donald Knuth, to get this information included in the next revision of TAOCP.
The most recent reference I am aware that covers this data lists 166 comparators as the best-known solution for a 29-element sorting network: Vinod K. Valsalam, Risto Miikkulainen, "Using Symmetry and Evolutionary Search to Minimize Sorting Networks", Journal of Machine Learning Research 14 (2013) 303-331