Best Known (174−23, 174, s)-Nets in Base 3
(174−23, 174, 16107)-Net over F3 — Constructive and digital
Digital (151, 174, 16107)-net over F3, using
- 32 times duplication [i] based on digital (149, 172, 16107)-net over F3, using
- net defined by OOA [i] based on linear OOA(3172, 16107, F3, 23, 23) (dual of [(16107, 23), 370289, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3172, 177178, F3, 23) (dual of [177178, 177006, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3172, 177186, F3, 23) (dual of [177186, 177014, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3166, 177147, F3, 23) (dual of [177147, 176981, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3172, 177186, F3, 23) (dual of [177186, 177014, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3172, 177178, F3, 23) (dual of [177178, 177006, 24]-code), using
- net defined by OOA [i] based on linear OOA(3172, 16107, F3, 23, 23) (dual of [(16107, 23), 370289, 24]-NRT-code), using
(174−23, 174, 59062)-Net over F3 — Digital
Digital (151, 174, 59062)-net over F3, using
- 32 times duplication [i] based on digital (149, 172, 59062)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3172, 59062, F3, 3, 23) (dual of [(59062, 3), 177014, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3172, 177186, F3, 23) (dual of [177186, 177014, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3166, 177147, F3, 23) (dual of [177147, 176981, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- OOA 3-folding [i] based on linear OA(3172, 177186, F3, 23) (dual of [177186, 177014, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3172, 59062, F3, 3, 23) (dual of [(59062, 3), 177014, 24]-NRT-code), using
(174−23, 174, large)-Net in Base 3 — Upper bound on s
There is no (151, 174, large)-net in base 3, because
- 21 times m-reduction [i] would yield (151, 153, large)-net in base 3, but