Best Known (150−21, 150, s)-Nets in Base 3
(150−21, 150, 5909)-Net over F3 — Constructive and digital
Digital (129, 150, 5909)-net over F3, using
- net defined by OOA [i] based on linear OOA(3150, 5909, F3, 21, 21) (dual of [(5909, 21), 123939, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3150, 59091, F3, 21) (dual of [59091, 58941, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, 59098, F3, 21) (dual of [59098, 58948, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(39, 49, F3, 4) (dual of [49, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3150, 59098, F3, 21) (dual of [59098, 58948, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3150, 59091, F3, 21) (dual of [59091, 58941, 22]-code), using
(150−21, 150, 29549)-Net over F3 — Digital
Digital (129, 150, 29549)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3150, 29549, F3, 2, 21) (dual of [(29549, 2), 58948, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3150, 59098, F3, 21) (dual of [59098, 58948, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(39, 49, F3, 4) (dual of [49, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(3150, 59098, F3, 21) (dual of [59098, 58948, 22]-code), using
(150−21, 150, large)-Net in Base 3 — Upper bound on s
There is no (129, 150, large)-net in base 3, because
- 19 times m-reduction [i] would yield (129, 131, large)-net in base 3, but