Best Known (96−23, 96, s)-Nets in Base 3
(96−23, 96, 400)-Net over F3 — Constructive and digital
Digital (73, 96, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 24, 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
(96−23, 96, 606)-Net over F3 — Digital
Digital (73, 96, 606)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(396, 606, F3, 23) (dual of [606, 510, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(396, 749, F3, 23) (dual of [749, 653, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [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(373, 729, F3, 19) (dual of [729, 656, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(396, 749, F3, 23) (dual of [749, 653, 24]-code), using
(96−23, 96, 32391)-Net in Base 3 — Upper bound on s
There is no (73, 96, 32392)-net in base 3, because
- 1 times m-reduction [i] would yield (73, 95, 32392)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2120 985678 555609 269296 672095 709809 221369 069985 > 395 [i]