Best Known (78, 102, s)-Nets in Base 3
(78, 102, 400)-Net over F3 — Constructive and digital
Digital (78, 102, 400)-net over F3, using
- 32 times duplication [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
(78, 102, 682)-Net over F3 — Digital
Digital (78, 102, 682)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3102, 682, F3, 24) (dual of [682, 580, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3102, 749, F3, 24) (dual of [749, 647, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(397, 729, F3, 25) (dual of [729, 632, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(379, 729, F3, 20) (dual of [729, 650, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3102, 749, F3, 24) (dual of [749, 647, 25]-code), using
(78, 102, 30039)-Net in Base 3 — Upper bound on s
There is no (78, 102, 30040)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4 638916 086267 232371 274155 566358 221123 727579 480257 > 3102 [i]