Best Known (179−35, 179, s)-Nets in Base 3
(179−35, 179, 688)-Net over F3 — Constructive and digital
Digital (144, 179, 688)-net over F3, using
- t-expansion [i] based on digital (142, 179, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (142, 180, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 45, 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, 45, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (142, 180, 688)-net over F3, using
(179−35, 179, 2339)-Net over F3 — Digital
Digital (144, 179, 2339)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3179, 2339, F3, 35) (dual of [2339, 2160, 36]-code), using
- 128 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 18 times 0, 1, 23 times 0, 1, 30 times 0) [i] based on linear OA(3162, 2194, F3, 35) (dual of [2194, 2032, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- 128 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 18 times 0, 1, 23 times 0, 1, 30 times 0) [i] based on linear OA(3162, 2194, F3, 35) (dual of [2194, 2032, 36]-code), using
(179−35, 179, 355332)-Net in Base 3 — Upper bound on s
There is no (144, 179, 355333)-net in base 3, because
- 1 times m-reduction [i] would yield (144, 178, 355333)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8 464249 417942 304149 539202 265223 665760 828489 393779 422490 114452 150142 198916 312365 142987 > 3178 [i]