The k-th Ding-Yuan matrix here is of order 3^(2k+1) and is a transform of A - I, where A is the adjacency matrix of the k-th in a family of strongly 3-regular graphs. The computation of interest is the rank modulo 3. The ranks are expected to be relatively small (rank nearer 2^(2k+1) when order is 3^(2k+1)). The graph family construction is defined in @article{DingYuan06, author="C. Ding and J. Yuan", title="A family of skew {H}adamard difference set", journal={J. Comb. Theory, Ser. A}, volume={113}, pages={1526--1535}, year=2006 }. Similar constructions (and the interest in rank modulo 3) are in @Article{Weng2007, author="Weng, Guobiao and Qiu, Weisheng and Wang, Zeying and Xiang, Qing", title="Pseudo-Paley graphs and skew Hadamard difference sets from presemifields", journal="Designs, Codes and Cryptography", year="2007", month="Sep", day="01", volume="44", number="1", pages="49--62", abstract="Let (K, + ,*) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, *) is a skew Hadamard difference set or a Paley type partial difference set in (K, +) according as q is congruent to 3 modulo 4 or q is congruent to 1 modulo 4. Applying this result to the Coulter--Matthews presemifield and the Ding--Yuan variation of it, we recover a recent construction of skew Hadamard difference sets by Ding and Yuan [7]. On the other hand, applying this result to the known presemifields with commutative multiplication and having order q congruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute the p-ranks of these pseudo-Paley graphs when q = 34, 36, 38, 310, 54, and 74. The p-rank results indicate that these graphs seem to be new. Along the way, we also disprove a conjecture of Ren{\'e} Peeters [17, p. 47] which says that the Paley graphs of nonprime order are uniquely determined by their parameters and the minimality of their relevant p-ranks.", issn="1573-7586", doi="10.1007/s10623-007-9057-6", url="https://doi.org/10.1007/s10623-007-9057-6" }