Best Known (135−18, 135, s)-Nets in Base 3
(135−18, 135, 19685)-Net over F3 — Constructive and digital
Digital (117, 135, 19685)-net over F3, using
- 31 times duplication [i] based on digital (116, 134, 19685)-net over F3, using
- net defined by OOA [i] based on linear OOA(3134, 19685, F3, 18, 18) (dual of [(19685, 18), 354196, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3134, 177165, F3, 18) (dual of [177165, 177031, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3134, 177170, F3, 18) (dual of [177170, 177036, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [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(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(31, 23, F3, 1) (dual of [23, 22, 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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3134, 177170, F3, 18) (dual of [177170, 177036, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3134, 177165, F3, 18) (dual of [177165, 177031, 19]-code), using
- net defined by OOA [i] based on linear OOA(3134, 19685, F3, 18, 18) (dual of [(19685, 18), 354196, 19]-NRT-code), using
(135−18, 135, 59057)-Net over F3 — Digital
Digital (117, 135, 59057)-net over F3, using
- 31 times duplication [i] based on digital (116, 134, 59057)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3134, 59057, F3, 3, 18) (dual of [(59057, 3), 177037, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3134, 177171, F3, 18) (dual of [177171, 177037, 19]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(15) [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(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(323, 24, F3, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,3)), using
- dual of repetition code with length 24 [i]
- linear OA(31, 24, F3, 1) (dual of [24, 23, 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 X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(3134, 177171, F3, 18) (dual of [177171, 177037, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3134, 59057, F3, 3, 18) (dual of [(59057, 3), 177037, 19]-NRT-code), using
(135−18, 135, large)-Net in Base 3 — Upper bound on s
There is no (117, 135, large)-net in base 3, because
- 16 times m-reduction [i] would yield (117, 119, large)-net in base 3, but