Best Known (76, 99, s)-Nets in Base 3
(76, 99, 400)-Net over F3 — Constructive and digital
Digital (76, 99, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (76, 100, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 25, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 25, 100)-net over F81, using
(76, 99, 712)-Net over F3 — Digital
Digital (76, 99, 712)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(399, 712, F3, 23) (dual of [712, 613, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(399, 757, F3, 23) (dual of [757, 658, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(391, 729, F3, 23) (dual of [729, 638, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(367, 729, F3, 17) (dual of [729, 662, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(399, 757, F3, 23) (dual of [757, 658, 24]-code), using
(76, 99, 43711)-Net in Base 3 — Upper bound on s
There is no (76, 99, 43712)-net in base 3, because
- 1 times m-reduction [i] would yield (76, 98, 43712)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 57270 235552 376952 615888 110389 136937 039719 254785 > 398 [i]