Best Known (81, 81+11, s)-Nets in Base 27
(81, 81+11, large)-Net over F27 — Constructive and digital
Digital (81, 92, large)-net over F27, using
- 271 times duplication [i] based on digital (80, 91, large)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 265722)-net over F27, using
- s-reduction based on digital (0, 0, s)-net over F27 with arbitrarily large s, using
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 0, 265722)-net over F27 (see above)
- digital (0, 1, 265722)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 265722)-net over F27 (see above)
- digital (0, 1, 265722)-net over F27 (see above)
- digital (0, 1, 265722)-net over F27 (see above)
- digital (0, 1, 265722)-net over F27 (see above)
- digital (0, 1, 265722)-net over F27 (see above)
- digital (3, 5, 265722)-net over F27, using
- s-reduction based on digital (3, 5, 551881)-net over F27, using
- digital (3, 5, 265722)-net over F27 (see above)
- digital (4, 7, 265722)-net over F27, using
- s-reduction based on digital (4, 7, 532899)-net over F27, using
- net defined by OOA [i] based on linear OOA(277, 532899, F27, 3, 3) (dual of [(532899, 3), 1598690, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(277, 532899, F27, 2, 3) (dual of [(532899, 2), 1065791, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(277, 532899, F27, 3, 3) (dual of [(532899, 3), 1598690, 4]-NRT-code), using
- s-reduction based on digital (4, 7, 532899)-net over F27, using
- digital (12, 17, 265722)-net over F27, using
- net defined by OOA [i] based on linear OOA(2717, 265722, F27, 5, 5) (dual of [(265722, 5), 1328593, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2717, 531445, F27, 5) (dual of [531445, 531428, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(2717, 531441, F27, 5) (dual of [531441, 531424, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2713, 531441, F27, 4) (dual of [531441, 531428, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(270, 4, F27, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(2717, 531445, F27, 5) (dual of [531445, 531428, 6]-code), using
- net defined by OOA [i] based on linear OOA(2717, 265722, F27, 5, 5) (dual of [(265722, 5), 1328593, 6]-NRT-code), using
- digital (40, 51, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (0, 0, 265722)-net over F27, using
- generalized (u, u+v)-construction [i] based on
(81, 81+11, large)-Net in Base 27 — Upper bound on s
There is no (81, 92, large)-net in base 27, because
- 9 times m-reduction [i] would yield (81, 83, large)-net in base 27, but