Best Known (165−23, 165, s)-Nets in Base 3
(165−23, 165, 5378)-Net over F3 — Constructive and digital
Digital (142, 165, 5378)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (128, 151, 5368)-net over F3, using
- net defined by OOA [i] based on linear OOA(3151, 5368, F3, 23, 23) (dual of [(5368, 23), 123313, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- OOA 11-folding and stacking with additional row [i] based on linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using
- net defined by OOA [i] based on linear OOA(3151, 5368, F3, 23, 23) (dual of [(5368, 23), 123313, 24]-NRT-code), using
- digital (3, 14, 10)-net over F3, using
(165−23, 165, 29551)-Net over F3 — Digital
Digital (142, 165, 29551)-net over F3, using
- 31 times duplication [i] based on digital (141, 164, 29551)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3164, 29551, F3, 2, 23) (dual of [(29551, 2), 58938, 24]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3162, 29550, F3, 2, 23) (dual of [(29550, 2), 58938, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3162, 59100, F3, 23) (dual of [59100, 58938, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3111, 59049, F3, 17) (dual of [59049, 58938, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(311, 51, F3, 5) (dual of [51, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(3162, 59100, F3, 23) (dual of [59100, 58938, 24]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3162, 29550, F3, 2, 23) (dual of [(29550, 2), 58938, 24]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3164, 29551, F3, 2, 23) (dual of [(29551, 2), 58938, 24]-NRT-code), using
(165−23, 165, large)-Net in Base 3 — Upper bound on s
There is no (142, 165, large)-net in base 3, because
- 21 times m-reduction [i] would yield (142, 144, large)-net in base 3, but