Best Known (127−22, 127, s)-Nets in Base 3
(127−22, 127, 1789)-Net over F3 — Constructive and digital
Digital (105, 127, 1789)-net over F3, using
- net defined by OOA [i] based on linear OOA(3127, 1789, F3, 22, 22) (dual of [(1789, 22), 39231, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3127, 19679, F3, 22) (dual of [19679, 19552, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3127, 19679, F3, 22) (dual of [19679, 19552, 23]-code), using
(127−22, 127, 6561)-Net over F3 — Digital
Digital (105, 127, 6561)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3127, 6561, F3, 3, 22) (dual of [(6561, 3), 19556, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- OOA 3-folding [i] based on linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using
(127−22, 127, 791699)-Net in Base 3 — Upper bound on s
There is no (105, 127, 791700)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 930109 822757 132863 694803 039666 539264 278957 694125 522908 616241 > 3127 [i]