Best Known (165−40, 165, s)-Nets in Base 3
(165−40, 165, 400)-Net over F3 — Constructive and digital
Digital (125, 165, 400)-net over F3, using
- 31 times duplication [i] based on digital (124, 164, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 41, 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, 41, 100)-net over F81, using
(165−40, 165, 824)-Net over F3 — Digital
Digital (125, 165, 824)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3165, 824, F3, 40) (dual of [824, 659, 41]-code), using
- 84 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 11 times 0, 1, 14 times 0, 1, 17 times 0, 1, 18 times 0) [i] based on linear OA(3154, 729, F3, 40) (dual of [729, 575, 41]-code), using
- an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 84 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 11 times 0, 1, 14 times 0, 1, 17 times 0, 1, 18 times 0) [i] based on linear OA(3154, 729, F3, 40) (dual of [729, 575, 41]-code), using
(165−40, 165, 35833)-Net in Base 3 — Upper bound on s
There is no (125, 165, 35834)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5 310132 709700 164421 894393 746618 406681 756937 913020 696726 231691 779301 342558 555977 > 3165 [i]