Best Known (157−37, 157, s)-Nets in Base 3
(157−37, 157, 464)-Net over F3 — Constructive and digital
Digital (120, 157, 464)-net over F3, using
- 31 times duplication [i] based on digital (119, 156, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 39, 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, 39, 116)-net over F81, using
(157−37, 157, 879)-Net over F3 — Digital
Digital (120, 157, 879)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3157, 879, F3, 37) (dual of [879, 722, 38]-code), using
- 135 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, 1, 21 times 0, 1, 23 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]
- 135 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, 1, 21 times 0, 1, 23 times 0) [i] based on linear OA(3142, 729, F3, 37) (dual of [729, 587, 38]-code), using
(157−37, 157, 51523)-Net in Base 3 — Upper bound on s
There is no (120, 157, 51524)-net in base 3, because
- 1 times m-reduction [i] would yield (120, 156, 51524)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 269 789776 717006 157463 547227 955209 021677 603789 905195 963501 666429 533635 150265 > 3156 [i]