J. R. Hendricks's digit(al) equations  

Acknowledgement: In J. R. Hendricks's self published "magic squares to tesseracts by computer" he explains the use of digit equations for the formation of Latin Hypercubes. Digital equation came to pass when he formulated his order 25 bimagic cube 

Note: the quality of the resulting hypercubes depend on parameters used the below give a description of the method and is not too concerned about the quality of te results, samples might be added at some future upload 

digit equation  modular equation to obtain latin hypercube  
LH(x) = [ _{k=0}∑^{n1} a_{k}x_{k} + a_{n} ] % m  
coordinate split  coordinate splitup in radix r when order m = r^{p}  
x_{k} = _{l=0}∑^{p1} x_{kp+l}r^{l}  
I currently don't know wether this can de done in compound orders other than those of single power, the database holds bimagic squares of orders 8 = 2^{3} and 9 = 3^{2} obtained by this method 

digital equation  modular equation to obtain latin hypercube on splitup coordinates  
LH(x) = [ _{k=0}∑^{n1}_{l=0}∑^{p1} a_{kp+l}x_{kp+l} + a_{np} ] % m  
Hypercube  
The hypercube is formed by combining latin hypercubes note (p = 1 for digit equations) 

H(a,x) = _{k=0}∑^{n1} m^{n1k} { [ ( _{l=0}∑^{n1}_{l=0}∑^{p1} a_{kp+l} x_{kp+l} + a_{np}) % m ]_{=[perm(0..m1)]} }  
the latin hypercubes digits obtained from the equation can be changed by means of applying a digitchanging permutation to those digits as indicated by _{=[perm(0..m1)]} (ie: replace digit d by perm(d) for all digits in the latin hypercube) 

Latin Prescription  
A matrix reformalism of the digit equations  
Latin Prescription  np+1 by n matrix representing the hypercube  
LP_{kp+i,j} = a_{kp+i,j} (i = 0..p1 ; k = 0..n1) (j = 0..n1) LP_{np,j} = a_{np,j} (j = 0..n1) 

note: j indicates that one needs n latin hypercubes / digit equations to formulate an hypercube; p is usually 1, if not digital equations are effectively described by the same formalism as digitequations 

H(LP) = _{k=0}∑^{n1} m^{n1k} { [ ( _{l=0}∑^{np} LP_{kp+l} x_{kp+l}) % m ]_{=[perm(0..m1)]} }  
in order for the above matrix multiplication to work note that x is a np+1 vector denoting the hypercubes coordinates (in normal or split up version) and an additional 1 as last entry and of course a digitchanging permutation _{=[perm(0..m1)]} can be performed at any latin hypercube individually. 

LP_{(k,i),j} = a_{k,i} (k = 0..p1 ; i = 0..n) (j = 0..n1)  
note: j indicates that one needs n latin hypercubes / digit equations to formulate an hypercube; p is usually 1, if not digital equations are effectively described by the same formalism as digitequations 

Examples  
LP_{i,j} = ((i == j1) ? 1 : 1); i,j = 0..n1; (ie n by n matrix all 1 except when i == j1 then 1) 

order 3 square: LP_{n=2,j} = {2,1} 7 0 5 2 4 6 3 8 1 
order 3 cube: LP_{n=3,j} = {1,0,0}  
09 25 05 23 00 16 07 14 18 
22 02 15 06 13 20 11 24 04 
08 12 19 10 26 03 21 01 17 

order 3 tesseract: LP_{n=4,j} = {0,2,2,2}  
cube 0 
26 36 58 30 79 11 64 05 51 
28 77 15 71 00 49 21 43 56 
66 07 47 19 41 60 35 72 13 
cube 1 
27 76 17 70 02 48 23 42 55 
68 06 46 18 40 62 34 74 12 
25 38 57 32 78 10 63 04 53 
cube 2 
67 08 45 20 39 61 33 73 14 
24 37 59 31 80 09 65 03 52 
29 75 16 69 01 50 22 44 54 