Best Known (77, 77+26, s)-Nets in Base 3
(77, 77+26, 252)-Net over F3 — Constructive and digital
Digital (77, 103, 252)-net over F3, using
- 31 times duplication [i] based on digital (76, 102, 252)-net over F3, using
- trace code for nets [i] based on digital (8, 34, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- trace code for nets [i] based on digital (8, 34, 84)-net over F27, using
(77, 77+26, 502)-Net over F3 — Digital
Digital (77, 103, 502)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3103, 502, F3, 26) (dual of [502, 399, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3103, 728, F3, 26) (dual of [728, 625, 27]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(3103, 728, F3, 26) (dual of [728, 625, 27]-code), using
(77, 77+26, 17073)-Net in Base 3 — Upper bound on s
There is no (77, 103, 17074)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13 918704 124278 593857 112850 031048 352927 124136 903725 > 3103 [i]