This is for the ring of integers Zv, where v is a not a power of a prime. The final column gives i values of m for up to s sets of treatments. There are other solutions for k = 2. To get a solution, the first initial block for the first set of treatments is given by the column headed block; the first initial block for a second or further sets of treatments is given by multiplying the elements of the block by the multiplier. The other initial blocks are given by multiplying these two blocks by coset representatives (for the first block) in the multiplicative group of the ring. Finally, develop these initial blocks additively.
| v | k | Block | Multiplier(s) |
| 65 | 2 | (1,64) | 8 |
| 85 | 2 | (1,84) | 13 |
| 91 | 2 | (1,90) | 9,10 |
| 133 | 2 | (1,132) | 11,12 |
| 145 | 2 | (1,144) | 12 |
| 185 | 2 | (1,184) | 43 |
| 205 | 2 | (1,204) | 32 |
| 217 | 2 | (1,216) | 25,26 |
| 221 | 2 | (1,220) | 21 |
| 247 | 2 | (1,246) | 68,69 |
| 259 | 2 | (1,258) | 100,101 |
| 265 | 2 | (1,264) | 23 |
| 301 | 2 | (1,300) | 79,80 |
| 305 | 2 | (1,304) | 72 |
| 325 | 2 | (1,324) | 18 |
| 341 | 2 | (1,340) | 29,47,159,163 |
| 365 | 2 | (1,364) | 27 |
| 377 | 2 | (1,376) | 70 |
| 403 | 2 | (1,402) | 87,88 |
| 425 | 2 | (1,424) | 132 |
| 427 | 2 | (1,426) | 74,75 |
| 445 | 2 | (1,444) | 123 |
| 451 | 2 | (1,450) | 59,107,127,174 |
| 469 | 2 | (1,468) | 37,38 |
| 481 | 2 | (1,480) | 31,162,193,269,270 |
| 493 | 2 | (1,492) | 157 |
| 91 | 3 | (1,9,81) | 10 |
| 91 | 3 | (1,16,74) | 17 |
| 133 | 3 | (1,11,121) | 12 |
| 133 | 3 | (1,30,102) | 31 |
| 217 | 3 | (1,37,431) | 38 |
| 217 | 3 | (1,163,305) | 164 |
| 247 | 3 | (1, 68, 178) | 69 |
| 247 | 3 | (1, 87, 159) | 88 |
| 259 | 3 | (1, 100, 158) | 101 |
| 259 | 3 | (1, 121, 137) | 122 |
| 301 | 3 | (1, 79, 221) | 80 |
| 301 | 3 | (1, 135, 165) | 136 |
| 403 | 3 | (1, 87, 315) | 88 |
| 403 | 3 | (1, 191, 211) | 192 |
| 427 | 3 | (1, 74, 352) | 75 |
| 427 | 3 | (1, 135, 291) | 136 |
| 469 | 3 | (1, 37, 431) | 38 |
| 469 | 3 | (1, 163, 305) | 164 |
| 481 | 3 | (1, 100, 380) | 45,101,216 |
| 481 | 3 | (1, 211, 269) | 31,193,212 |
| 481 | 4 | (1,31,450,480) | 211,162 |
| 481 | 4 | (1, 216, 480, 265) | 97,119 |
| 341 | 5 | (1, 4, 16, 64, 256) | 85 |
| 341 | 5 | (1, 47, 159, 163, 312) | 29 |
| 341 | 5 | (1, 70, 126, 190, 295) | 46 |
| 341 | 5 | (1, 97, 157, 202, 225) | 116 |
| 451 | 5 | (1, 16, 37, 141, 256) | 195 |
| 451 | 5 | (1, 201, 92, 262, 346) | 105 |
| 451 | 5 | (1, 119, 180, 223, 379) | 72 |
| 451 | 5 | (1, 59, 174, 324, 344) | 107 |
| 481 | 6 | (1, 100, 101, 381, 380, 480) | 45 |
The solution given by Furino for k = 3 was for numbers of the form r2+r+1, where r = 0,2 mod 3, The fundamental subgroup was of form {1,r,r2}. Only two of the solutions given above are of this form (v = 91, 133). All of them can be solved as Galois Rings, as can all the solutions for other k. There also values of v for which there is no solution here, but for which a Galois Ring solution would exist e.g. for k = 2, 45 and other multiples of 9.
The solutions for k = 2 are all equivalent to SAMDRRs or SOLS. There are therefore, for example, 5 mutually orthogonal SOLS for 481. In all cases, the first block is of the form (1,-1). I assume this has been done elsewhere at some time.