Best Known (142−34, 142, s)-Nets in Base 3
(142−34, 142, 400)-Net over F3 — Constructive and digital
Digital (108, 142, 400)-net over F3, using
- 32 times duplication [i] based on digital (106, 140, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 35, 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, 35, 100)-net over F81, using
(142−34, 142, 781)-Net over F3 — Digital
Digital (108, 142, 781)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3142, 781, F3, 34) (dual of [781, 639, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3142, 785, F3, 34) (dual of [785, 643, 35]-code), using
- 44 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 9 times 0, 1, 11 times 0) [i] based on linear OA(3130, 729, F3, 34) (dual of [729, 599, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 44 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 9 times 0, 1, 11 times 0) [i] based on linear OA(3130, 729, F3, 34) (dual of [729, 599, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3142, 785, F3, 34) (dual of [785, 643, 35]-code), using
(142−34, 142, 34679)-Net in Base 3 — Upper bound on s
There is no (108, 142, 34680)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 56 402332 236318 514178 975644 028920 817077 148877 142090 727261 966396 140017 > 3142 [i]