Best Known (127−23, 127, s)-Nets in Base 3
(127−23, 127, 688)-Net over F3 — Constructive and digital
Digital (104, 127, 688)-net over F3, using
- t-expansion [i] based on digital (103, 127, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (103, 128, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 32, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 32, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (103, 128, 688)-net over F3, using
(127−23, 127, 3295)-Net over F3 — Digital
Digital (104, 127, 3295)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3127, 3295, F3, 2, 23) (dual of [(3295, 2), 6463, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3127, 6590, F3, 23) (dual of [6590, 6463, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 6591, F3, 23) (dual of [6591, 6464, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3127, 6591, F3, 23) (dual of [6591, 6464, 24]-code), using
- OOA 2-folding [i] based on linear OA(3127, 6590, F3, 23) (dual of [6590, 6463, 24]-code), using
(127−23, 127, 716448)-Net in Base 3 — Upper bound on s
There is no (104, 127, 716449)-net in base 3, because
- 1 times m-reduction [i] would yield (104, 126, 716449)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 310033 302082 546322 194902 998847 980161 377928 977617 317770 818939 > 3126 [i]