Best Known (157, 194, s)-Nets in Base 3
(157, 194, 692)-Net over F3 — Constructive and digital
Digital (157, 194, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 18, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (139, 176, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 44, 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, 44, 172)-net over F81, using
- digital (0, 18, 4)-net over F3, using
(157, 194, 3285)-Net over F3 — Digital
Digital (157, 194, 3285)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3194, 3285, F3, 2, 37) (dual of [(3285, 2), 6376, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3194, 6570, F3, 37) (dual of [6570, 6376, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3185, 6561, F3, 35) (dual of [6561, 6376, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- OOA 2-folding [i] based on linear OA(3194, 6570, F3, 37) (dual of [6570, 6376, 38]-code), using
(157, 194, 493042)-Net in Base 3 — Upper bound on s
There is no (157, 194, 493043)-net in base 3, because
- 1 times m-reduction [i] would yield (157, 193, 493043)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 121 451451 519680 736279 999999 124185 016653 855708 472256 593219 170140 947002 768529 405488 723042 182245 > 3193 [i]