Best Known (118−27, 118, s)-Nets in Base 3
(118−27, 118, 464)-Net over F3 — Constructive and digital
Digital (91, 118, 464)-net over F3, using
- 32 times duplication [i] based on digital (89, 116, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 29, 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, 29, 116)-net over F81, using
(118−27, 118, 803)-Net over F3 — Digital
Digital (91, 118, 803)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3118, 803, F3, 27) (dual of [803, 685, 28]-code), using
- 65 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0) [i] based on linear OA(3108, 728, F3, 27) (dual of [728, 620, 28]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 65 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0) [i] based on linear OA(3108, 728, F3, 27) (dual of [728, 620, 28]-code), using
(118−27, 118, 55766)-Net in Base 3 — Upper bound on s
There is no (91, 118, 55767)-net in base 3, because
- 1 times m-reduction [i] would yield (91, 117, 55767)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 66 570149 039528 914751 328661 773851 099671 551356 507215 188567 > 3117 [i]