Identification | ||
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Knight Jump vectors (odd orders) |
KJ(P,V_{0} .. V_{n-1}) | |
The knightjump construction consist of the position of the first number and a total of n vectors which point from one number onto the nxxt |
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In the regular number rang P is the postion of the number '1', V_{k} k = 0 .. n-1, is the vector between the number m^{k} to m^{k} + 1 |
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Latin Prescription vectors (any orders) |
lp( a_{j} ) _{=P[0..m-1]} (for single component) | |
The latin prescription construction consist of a series of modular equations one for each component. Aside from these each component can be subject to digit changing. (The Hendricks digit and digital equation belong also to this method thuugh he uses coordinate starting with '1' in stead of '0') |
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due to other subject I'm currently out of touch with the method, so other then a trial and error based process I currently can't provide. |
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pandiagonal construction (prime orders) |
LH( a_{k} )_{=P[0..m-1]} (for single component) | |
The pandiagonal construction for prime order hypercubes consist of n parameter n-Vectors, also digit changing permutation can be applied to each component |
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when digit changing is applied identification is a bit complicated, thus far the cubes i've seen where without, and so fairly simple to recognise take a number at a certain position and write it into m-based numbers do the same with the numbers next to it along each of the n 1-agonals each difference in the digits defines a number in the vector. |
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These qualitative description give the process of reobtaining LH(a_{j})_{=P[0..m-1]} change hypercube to analitic numberrange, and normalized position (optional) write the numbers in radix m notation, each digit combine into a latin hypercube (LH) with each hypercube suppose the poitive difference of a number with the next along a 1-agonal direction is the disired parameter a_{j} these parameters form LH(a_{j}) When LH(a_{j}) reconstructs the given hypercube the hypercube is found Positive difference of d1 and d2: dif(d1,d2) = (m + d1 - d2) % m ; thus d2 = (d1 + dif(d1,d2)) % m note that one might construct the difference hypercube, this should show constant features (I haven't done this yet, exact feature I'll verify) (NOTE: thus far I've seen no cube wher I needed to go further then this.) If LH(a_{j}) does not reconstruct the given hypercube digit changing might be involved I suppose with the following one might obtain the parameters and permutation suppose the parameter of the first 1-agonal and construct the 1 agonal line, this line with the hypercubes 1-agonal will give the hypercube digit changing permutation P[0..m-1] Using this permutaton in reverse the other parameters of the latin hypercube can be obtained The thus found numbers form LH(a_{j})_{=P[0..m-1]} Note:it might well be possible to use diferent permutation in the various 1-agonal directions (this generalisation of the construction method needs to be investigated yet) |