How would I generate matrices to represent the variants of "R" in these equations?

I am essentially trying to make my own code for the nonpartest() function in the npmv package. I have a dataset:

Cattle <- read.table(text=" Treatment Replicate Weight_Loss Persistent Head_Size Salebarn_Q
                             'LA 200'         1       17.90      14.10     14.25        1.0
                             'LA 200'         2       19.30      15.30      2.56        1.0
                             'LA 200'         3       19.50      16.82      5.80        1.5
                             'LA 200'         4       18.94      12.70      7.51        1.5
                               Excede         1       19.60      11.20     14.52        1.0
                               Excede         2       19.50      10.54      9.83        1.0
                               Excede         3       19.10      10.83      3.82        0.5
                               Excede         4       20.40      11.00      0.04        1.0
                              Micotil         1       17.30      14.29      1.62        1.0
                              Micotil         2       20.00      11.65      0.13        3.0
                              Micotil         3       18.10      10.89      2.41        0.0
                              Micotil         4       19.50      12.43      5.93        2.0
                               Zoetis         1       18.50      25.48     10.08        1.0
                               Zoetis         2       17.60      20.12     11.93        1.0
                               Zoetis         3       19.70      23.29      7.93        2.5
                               Zoetis         4       18.50      28.32     13.08        3.0", header=TRUE)

Which I am trying to use to generate the matrices for Ri. and R.. and Rij in the equation in the paper below so that I can calculate the test statistics G and H

I attempted to do it using

R<-matrix(rank(Cattle,ties.method = "average"),N,p)
R_bar<-matrix(rank(Cattle,ties.method = "average"),1,p)
H<-(1/(a-1))*sum(n*(R-R_bar)*t(R-R_bar))
G<-(1/(N-a)*sum(sum(R-R_bar)*(R_prime-R_bar_prime)))

But that does not work apparently.. I'm not entirely sure what they're describing in the paper in regards to the dimensions of the R matrices.. I know you should use the rank() function and then transpose them using t() for the 'prime' versions

**Images show the excerpts of the paper where the different matrices and their dimensions and how they go in the actual equations are described