Best Known (154, 154+37, s)-Nets in Base 3
(154, 154+37, 688)-Net over F3 — Constructive and digital
Digital (154, 191, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (154, 196, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 49, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 49, 172)-net over F81, using
(154, 154+37, 2488)-Net over F3 — Digital
Digital (154, 191, 2488)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3191, 2488, F3, 37) (dual of [2488, 2297, 38]-code), using
- 278 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0, 1, 30 times 0, 1, 37 times 0, 1, 44 times 0, 1, 51 times 0) [i] based on linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 278 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0, 1, 30 times 0, 1, 37 times 0, 1, 44 times 0, 1, 51 times 0) [i] based on linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using
(154, 154+37, 410545)-Net in Base 3 — Upper bound on s
There is no (154, 191, 410546)-net in base 3, because
- 1 times m-reduction [i] would yield (154, 190, 410546)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 498264 224856 314078 579573 519050 518123 493291 248263 490724 471979 729549 124006 116396 829381 847637 > 3190 [i]