Best Known (134, 134+15, s)-Nets in Base 3
(134, 134+15, 683290)-Net over F3 — Constructive and digital
Digital (134, 149, 683290)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 9)-net over F3, using
- digital (125, 140, 683281)-net over F3, using
- net defined by OOA [i] based on linear OOA(3140, 683281, F3, 15, 15) (dual of [(683281, 15), 10249075, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3140, 4782968, F3, 15) (dual of [4782968, 4782828, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(3140, 4782968, F3, 15) (dual of [4782968, 4782828, 16]-code), using
- net defined by OOA [i] based on linear OOA(3140, 683281, F3, 15, 15) (dual of [(683281, 15), 10249075, 16]-NRT-code), using
(134, 134+15, 1849542)-Net over F3 — Digital
Digital (134, 149, 1849542)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3149, 1849542, F3, 2, 15) (dual of [(1849542, 2), 3698935, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3149, 2391509, F3, 2, 15) (dual of [(2391509, 2), 4782869, 16]-NRT-code), using
- 1 step truncation [i] based on linear OOA(3150, 2391510, F3, 2, 16) (dual of [(2391510, 2), 4782870, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3150, 4783020, F3, 16) (dual of [4783020, 4782870, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(3141, 4782969, F3, 16) (dual of [4782969, 4782828, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(399, 4782969, F3, 11) (dual of [4782969, 4782870, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(39, 51, F3, 4) (dual of [51, 42, 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(15) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(3150, 4783020, F3, 16) (dual of [4783020, 4782870, 17]-code), using
- 1 step truncation [i] based on linear OOA(3150, 2391510, F3, 2, 16) (dual of [(2391510, 2), 4782870, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3149, 2391509, F3, 2, 15) (dual of [(2391509, 2), 4782869, 16]-NRT-code), using
(134, 134+15, large)-Net in Base 3 — Upper bound on s
There is no (134, 149, large)-net in base 3, because
- 13 times m-reduction [i] would yield (134, 136, large)-net in base 3, but