Best Known (118, 118+37, s)-Nets in Base 3
(118, 118+37, 400)-Net over F3 — Constructive and digital
Digital (118, 155, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (118, 156, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 39, 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, 39, 100)-net over F81, using
(118, 118+37, 831)-Net over F3 — Digital
Digital (118, 155, 831)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3155, 831, F3, 37) (dual of [831, 676, 38]-code), using
- 89 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 13 times 0, 1, 17 times 0, 1, 19 times 0) [i] based on linear OA(3142, 729, F3, 37) (dual of [729, 587, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 89 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 13 times 0, 1, 17 times 0, 1, 19 times 0) [i] based on linear OA(3142, 729, F3, 37) (dual of [729, 587, 38]-code), using
(118, 118+37, 45600)-Net in Base 3 — Upper bound on s
There is no (118, 155, 45601)-net in base 3, because
- 1 times m-reduction [i] would yield (118, 154, 45601)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 29 973429 547031 845560 230384 015084 657555 721207 885032 316523 675889 336486 584969 > 3154 [i]