Best Known (131, 131+20, s)-Nets in Base 3
(131, 131+20, 17718)-Net over F3 — Constructive and digital
Digital (131, 151, 17718)-net over F3, using
- 31 times duplication [i] based on digital (130, 150, 17718)-net over F3, using
- net defined by OOA [i] based on linear OOA(3150, 17718, F3, 20, 20) (dual of [(17718, 20), 354210, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3150, 177180, F3, 20) (dual of [177180, 177030, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, 177186, F3, 20) (dual of [177186, 177036, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3111, 177147, F3, 16) (dual of [177147, 177036, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(19) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3150, 177186, F3, 20) (dual of [177186, 177036, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3150, 177180, F3, 20) (dual of [177180, 177030, 21]-code), using
- net defined by OOA [i] based on linear OOA(3150, 17718, F3, 20, 20) (dual of [(17718, 20), 354210, 21]-NRT-code), using
(131, 131+20, 59062)-Net over F3 — Digital
Digital (131, 151, 59062)-net over F3, using
- 31 times duplication [i] based on digital (130, 150, 59062)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3150, 59062, F3, 3, 20) (dual of [(59062, 3), 177036, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3150, 177186, F3, 20) (dual of [177186, 177036, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3111, 177147, F3, 16) (dual of [177147, 177036, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(19) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(3150, 177186, F3, 20) (dual of [177186, 177036, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3150, 59062, F3, 3, 20) (dual of [(59062, 3), 177036, 21]-NRT-code), using
(131, 131+20, large)-Net in Base 3 — Upper bound on s
There is no (131, 151, large)-net in base 3, because
- 18 times m-reduction [i] would yield (131, 133, large)-net in base 3, but