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"
}