Best Known (73−23, 73, s)-Nets in Base 27
(73−23, 73, 1791)-Net over F27 — Constructive and digital
Digital (50, 73, 1791)-net over F27, using
- 272 times duplication [i] based on digital (48, 71, 1791)-net over F27, using
- net defined by OOA [i] based on linear OOA(2771, 1791, F27, 23, 23) (dual of [(1791, 23), 41122, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2771, 19702, F27, 23) (dual of [19702, 19631, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(2767, 19683, F27, 23) (dual of [19683, 19616, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2752, 19683, F27, 18) (dual of [19683, 19631, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(274, 19, F27, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(2771, 19702, F27, 23) (dual of [19702, 19631, 24]-code), using
- net defined by OOA [i] based on linear OOA(2771, 1791, F27, 23, 23) (dual of [(1791, 23), 41122, 24]-NRT-code), using
(73−23, 73, 19710)-Net over F27 — Digital
Digital (50, 73, 19710)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2773, 19710, F27, 23) (dual of [19710, 19637, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(15) [i] based on
- linear OA(2767, 19683, F27, 23) (dual of [19683, 19616, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2746, 19683, F27, 16) (dual of [19683, 19637, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- Reed–Solomon code RS(21,27) [i]
- construction X applied to Ce(22) ⊂ Ce(15) [i] based on
(73−23, 73, large)-Net in Base 27 — Upper bound on s
There is no (50, 73, large)-net in base 27, because
- 21 times m-reduction [i] would yield (50, 52, large)-net in base 27, but