Best Known (135−31, 135, s)-Nets in Base 3
(135−31, 135, 464)-Net over F3 — Constructive and digital
Digital (104, 135, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (104, 136, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 34, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 34, 116)-net over F81, using
(135−31, 135, 868)-Net over F3 — Digital
Digital (104, 135, 868)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3135, 868, F3, 31) (dual of [868, 733, 32]-code), using
- 122 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 17 times 0, 1, 20 times 0, 1, 23 times 0) [i] based on linear OA(3118, 729, F3, 31) (dual of [729, 611, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 122 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 17 times 0, 1, 20 times 0, 1, 23 times 0) [i] based on linear OA(3118, 729, F3, 31) (dual of [729, 611, 32]-code), using
(135−31, 135, 58737)-Net in Base 3 — Upper bound on s
There is no (104, 135, 58738)-net in base 3, because
- 1 times m-reduction [i] would yield (104, 134, 58738)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8596 970275 188739 620826 434675 749776 296443 211761 025261 348330 439641 > 3134 [i]