Best Known (143−19, 143, s)-Nets in Base 3
(143−19, 143, 19690)-Net over F3 — Constructive and digital
Digital (124, 143, 19690)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (114, 133, 19683)-net over F3, using
- net defined by OOA [i] based on linear OOA(3133, 19683, F3, 19, 19) (dual of [(19683, 19), 373844, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using
- net defined by OOA [i] based on linear OOA(3133, 19683, F3, 19, 19) (dual of [(19683, 19), 373844, 20]-NRT-code), using
- digital (1, 10, 7)-net over F3, using
(143−19, 143, 59063)-Net over F3 — Digital
Digital (124, 143, 59063)-net over F3, using
- 31 times duplication [i] based on digital (123, 142, 59063)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3142, 59063, F3, 3, 19) (dual of [(59063, 3), 177047, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3142, 177189, F3, 19) (dual of [177189, 177047, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3141, 177188, F3, 19) (dual of [177188, 177047, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3100, 177147, F3, 14) (dual of [177147, 177047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3141, 177188, F3, 19) (dual of [177188, 177047, 20]-code), using
- OOA 3-folding [i] based on linear OA(3142, 177189, F3, 19) (dual of [177189, 177047, 20]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3142, 59063, F3, 3, 19) (dual of [(59063, 3), 177047, 20]-NRT-code), using
(143−19, 143, large)-Net in Base 3 — Upper bound on s
There is no (124, 143, large)-net in base 3, because
- 17 times m-reduction [i] would yield (124, 126, large)-net in base 3, but